[wohoromh] factoring binary square root of half

consider the irrational number sqrt(0.5) = sqrt(2)/2 = sin(pi/4) = cos(pi/4) ~= 0.707 , expressed in binary: 0.1011010100000100111100110011001111111001110111100110010010000100010...  take the first N bits past the decimal point (radix point), and interpret as an integer.  if the number is even, add one (designated "ceil" in the table below), because trailing zeros simply multiply by powers of 2.  we compute the integer factorization for various N.

the pattern of ceil and floor match the binary digits 0 and 1.

in comparison, the "2^n-1" Cunningham project factors integers produced by 0.11111... in binary.

unlike Cunningham, no interesting patterns are readily apparent.  perhaps this data will be useful for someone else to examine more carefully.

the largest prime found was at N = 278.  it would be very easy to search for larger.

CPU limit of approximately 10000 seconds per number, though did retry with more CPU for a few numbers which were almost done with MPQS.  it would be nice if were easy to direct Pari/GP not to try MPQS for numbers larger than a user-given threshold.  (it's possible with successive calls to factorint with various flags, not explored here.)  (the "alarm" function might also be useful.)

previously, factoring Newton's method.

future work: 1/e, log 2, 1/phi.  we prefer numbers between 0 and 1 because it doesn't seem contrived to discard the integer part.

scripts using Pari/GP, GNU Parallel, Perl, and Bash.

for(i=1, +oo, x=floor(sqrt(0.5)*2^i); print1(i); if(x%2, print(" floor ",factorint(x)), print(" ceil ",factorint(x+1))))

1 floor matrix(0,2)
2 ceil Mat([3, 1])
3 floor Mat([5, 1])
4 floor Mat([11, 1])
5 ceil Mat([23, 1])
6 floor [3, 2; 5, 1]
7 ceil [7, 1; 13, 1]
8 floor Mat([181, 1])
9 ceil [3, 1; 11, 2]
10 ceil [5, 2; 29, 1]
11 ceil [3, 2; 7, 1; 23, 1]
12 ceil Mat([2897, 1])
13 ceil [3, 1; 1931, 1]
14 floor [5, 1; 7, 1; 331, 1]
15 ceil [17, 1; 29, 1; 47, 1]
16 ceil [3, 2; 19, 1; 271, 1]
17 floor Mat([92681, 1])
18 floor Mat([185363, 1])
19 floor [7, 1; 211, 1; 251, 1]
20 floor [5, 1; 11, 1; 13, 1; 17, 1; 61, 1]
21 ceil [67, 1; 22133, 1]
22 ceil [3, 1; 988607, 1]
23 floor Mat([5931641, 1])
24 floor [181, 1; 65543, 1]
25 ceil [71, 1; 334177, 1]
26 ceil [3, 1; 7, 1; 13, 1; 101, 1; 1721, 1]
27 floor [5, 1; 683, 1; 27791, 1]
28 floor [17, 1; 11165443, 1]
29 ceil [47, 1; 8077129, 1]
30 ceil [3, 3; 5, 3; 23, 1; 9781, 1]
31 floor [7, 2; 31, 1; 999671, 1]
32 floor [13, 1; 233615423, 1]
33 floor [11, 1; 97, 1; 5692597, 1]
34 floor [7, 1; 29137, 1; 59561, 1]
35 floor [9851, 1; 2466349, 1]
36 floor [31, 1; 239, 1; 6558511, 1]
37 floor [7, 1; 37, 1; 9883, 1; 37967, 1]
38 ceil [17, 1; 19, 1; 29, 1; 227, 1; 91411, 1]
39 ceil [3, 1; 1301, 1; 99599299, 1]
40 floor [11, 1; 13487, 1; 5240549, 1]
41 floor [35603, 1; 43674529, 1]
42 floor [5, 2; 7, 1; 2621, 1; 6780157, 1]
43 ceil [17, 1; 1151, 1; 2383, 1; 133391, 1]
44 floor [3, 2; 224797, 1; 6148537, 1]
45 floor [13, 1; 1913777545831, 1]
46 floor [3, 1; 7, 1; 11, 1; 1361, 1; 10847, 1; 14591, 1]
47 floor [5, 1; 61, 1; 101, 1; 947, 1; 3411329, 1]
48 ceil [3, 1; 66344288255477, 1]
49 ceil [13, 1; 31, 1; 9173, 1; 107680819, 1]
50 floor [3, 1; 23, 1; 4937, 1; 2337074557, 1]
51 floor [149, 1; 10686328309607, 1]
52 ceil [3, 1; 29, 1; 1831, 1; 8609, 1; 2322119, 1]
53 ceil [7, 2; 11, 1; 17, 1; 6173, 1; 112600627, 1]
54 floor [3, 1; 5, 1; 2273, 1; 373606198711, 1]
55 ceil [13, 2; 37, 1; 4074237436447, 1]
56 ceil [101, 1; 9463, 1; 53419, 1; 997973, 1]
57 floor [7, 1; 8487329, 1; 1715243087, 1]
58 ceil [3, 1; 28559, 1; 2378814075199, 1]
59 ceil [5, 1; 4153, 1; 19630113510313, 1]
60 ceil [3, 1; 11, 1; 45827, 1; 539075226979, 1]
61 ceil [503838241, 1; 3236112497, 1]
62 floor [3, 3; 991, 1; 283183, 1; 430368163, 1]
63 ceil [151, 1; 43191449752757557, 1]
64 ceil [11, 2; 107800147316799853, 1]
65 ceil [5, 2; 547, 1; 1907688164582491, 1]
66 floor [74771, 1; 830639, 1; 840077221, 1]
67 ceil [3, 1; 7, 1; 719, 1; 6911089648497401, 1]
68 floor [11, 1; 229, 1; 82850768243479363, 1]
69 floor [5, 1; 12329, 1; 75598937, 1; 89565583, 1]
70 ceil [13, 1; 109, 1; 589135032336837023, 1]
71 ceil [3, 2; 29, 1; 43, 1; 6056959, 1; 24561285733, 1]
72 floor [7, 1; 743, 1; 2203, 1; 2753, 1; 156089, 1; 678211, 1]
73 ceil [61, 1; 163, 1; 709, 1; 3671, 1; 11813, 1; 21845723, 1]
74 floor [3, 1; 5, 1; 443, 1; 1907, 1; 2473, 1; 426221042131, 1]
75 floor [61553089349, 1; 433995095759, 1]
76 floor [3, 4; 7, 1; 3805933, 1; 24758280277253, 1]
77 floor Mat([106854955625126151883567, 1])
78 floor [3, 1; 5, 1; 71, 1; 10578400013, 1; 18969464483, 1]
79 ceil [7, 1; 23, 1; 790271, 1; 3359330541313441, 1]
80 floor [1361, 1; 520100201, 1; 1207645614181, 1]
81 floor [3, 3; 29, 1; 2183498454664135926101, 1]
82 ceil [37, 1; 3643, 1; 7253, 1; 3497567251105229, 1]
83 ceil [11, 1; 1269743, 1; 338115209, 1; 1448109569, 1]
84 ceil [5, 1; 7, 1; 137, 1; 54767, 1; 52083129834398261, 1]
85 floor [1899051391, 1; 14404490984115919, 1]
86 ceil [3, 1; 229, 1; 29837, 1; 1396517, 1; 1911201647533, 1]
87 ceil [11, 1; 53, 1; 79, 1; 157007, 1; 26335763, 1; 574558241, 1]
88 floor [3, 2; 7, 2; 173, 1; 20054365103, 1; 143031275227, 1]
89 floor [60661, 1; 4386530623, 1; 1644840857689, 1]
90 ceil [3, 1; 5, 1; 29871311, 1; 918371557, 1; 2127260347, 1]
91 floor [765199, 1; 1246033, 1; 1836160622392307, 1]
92 floor [13, 1; 25297739, 1; 10646810968801884277, 1]
93 ceil [17, 1; 59, 1; 6485489, 1; 1076541902983530037, 1]
94 ceil [3, 3; 263, 1; 52433, 1; 89669, 1; 419505876135341, 1]
95 floor [11579, 1; 19927, 1; 121400818694223508861, 1]
96 floor [11, 1; 3691, 1; 1379837220137093665642027, 1]
97 ceil [5, 1; 47, 1; 4363, 1; 28099, 1; 3889113716802424469, 1]
98 floor [3, 1; 13, 1; 311, 1; 36083, 1; 512031810562446000287, 1]
99 floor [7, 1; 666303759607, 1; 96091344297603931, 1]
100 floor [3, 2; 19, 1; 1907, 1; 23497, 1; 328787, 1; 2115121, 1; 168219173, 1]
101 ceil [23, 1; 79, 1; 1061, 1; 8966311, 1; 103712345123491597, 1]
102 floor [13, 1; 344988449034193, 1; 799459841988673, 1]
103 ceil [3, 1; 5, 1; 461, 1; 13955057, 1; 2311769177, 1; 32144471329, 1]
104 floor [7, 1; 149, 1; 193, 1; 71246401470177456515906071, 1]
105 ceil [1697, 1; 7001, 1; 525402437411, 1; 4595159004577, 1]
106 floor [3, 1; 9725489, 1; 1966218784412143090415351, 1]
107 ceil [5, 1; 1553, 1; 77317, 1; 191107655075397676546867, 1]
108 ceil [61, 1; 523, 1; 7653761, 1; 60887839, 1; 15434321868437, 1]
109 floor [436996140548609, 1; 1050211883448884393, 1]
110 ceil [3, 1; 5, 2; 283, 1; 8227127, 1; 341853091, 1; 15376203344479, 1]
111 floor [139, 1; 658477, 1; 557303963, 1; 35988767956229041, 1]
112 ceil [17, 1; 215971077564916733277295076686747, 1]
113 floor [3, 2; 149, 1; 40841, 1; 16300385159, 1; 8225295336318343, 1]
114 ceil [5, 1; 1049, 1; 4890563, 1; 37378068001, 1; 15317338418957, 1]
115 floor [113, 1; 929, 1; 49823, 1; 463291, 1; 504521, 1; 24025772497369, 1]
116 floor [3, 1; 13, 1; 67, 1; 22481489895773957692852759609183, 1]
117 floor Mat([117488266195314702902848521717590359, 1])
118 floor [3, 1; 11, 1; 17, 1; 29, 1; 1588183, 1; 9833748901, 1; 924791807365897, 1]
119 floor [158243, 1; 209837, 1; 640384439, 1; 22100756494859711, 1]
120 ceil [3, 2; 1902427, 1; 2019763147397, 1; 27179003377242049, 1]
121 floor [7, 1; 79, 1; 1598507, 1; 2126546109021791137930599967, 1]
122 ceil [5, 1; 11, 1; 19087340017456141, 1; 3581264301899243153, 1]
123 ceil [3, 1; 13, 1; 9649, 1; 251298847, 1; 79512802243629107009837, 1]
124 floor Mat([15038498073000281971564610779851566057, 1])
125 floor [5, 1; 47, 1; 45631, 1; 52291, 1; 53638875454326506590530029, 1]
126 floor [7, 1; 23, 1; 67712237018159551, 1; 5517869409876019321, 1]
127 floor [97, 1; 832327, 1; 1987937787637, 1; 749593729871700821, 1]
128 floor [11, 3; 1973, 1; 92749067, 1; 323681864119, 1; 3052048215973, 1]
129 ceil [5, 1; 7, 1; 17, 1; 14168007299, 1; 57085880646818316174639991, 1]
130 ceil [3, 2; 25074727, 1; 2868525614417, 1; 1486780934779445939, 1]
131 ceil [1165667, 1; 9182480030607967, 1; 179837368820291053, 1]
132 floor [3, 1; 7, 1; 19, 1; 1723, 1; 5743, 1; 897284722951, 1; 1086718766211937253, 1]
133 floor Mat([7699711013376144369441080719284001821667, 1])
134 floor [3, 3; 5, 1; 752991731, 1; 31397252659631, 1; 4824905345129461, 1]
135 ceil [7, 1; 4399834864786368211109188982448001040953, 1]
136 floor [37, 1; 233, 1; 5058409, 1; 201873653, 1; 6068813461, 1; 1152947137393, 1]
137 ceil [3, 1; 1277, 1; 32157498359179929499101355131439318493, 1]
138 floor [5, 1; 47, 1; 239, 1; 474278573, 1; 9249646515921324285371487797, 1]
139 floor [59, 1; 109, 1; 131, 1; 9871, 1; 59257515981411614899527996908201, 1]
140 ceil [93319344496781774179, 1; 10561186590269445823997, 1]
141 floor [3, 2; 5, 2; 709308931063, 1; 12350838545344761912566131931, 1]
142 floor [79, 1; 4113692265733, 1; 53003955241997, 1; 228863858932469, 1]
143 floor [3, 1; 439, 1; 4009631, 1; 46320102226711, 1; 32234036321265935699, 1]
144 floor [11, 1; 19, 1; 37, 1; 47, 1; 5884677943018451957, 1; 7372856987937987401, 1]
145 ceil [3, 1; 5, 1; 12917, 1; 19191104951, 1; 24899787397, 1; 340632306591291679, 1]
146 floor [349, 1; 257053, 1; 50763052642139, 1; 13850598329656730918863, 1]
147 floor [67, 1; 71, 1; 1669, 1; 3049, 1; 3274697, 1; 1591388921446543439864325491, 1]
148 ceil [83, 1; 409, 1; 501255697, 1; 17329159939997, 1; 855630246309561353, 1]
149 ceil [3, 1; 7, 1; 4285709075147, 1; 5606765271951119958360877803851, 1]
150 ceil [2521, 1; 13469, 1; 128521, 1; 9387408898949, 1; 24635213006982012713, 1]
151 ceil [3, 2; 5, 1; 13, 1; 29, 2; 1193, 1; 1933, 1; 122572526113, 1; 14514331944621456694781, 1]
152 ceil [7, 1; 277, 1; 1827402277, 1; 1899402623, 1; 47916644683, 1; 12517826835907, 1]
153 floor [3, 1; 137, 1; 163, 1; 12056097430753379, 1; 9996274269421736110084091, 1]
154 ceil [5, 1; 3539, 1; 912543902295778915889352998734551171987229, 1]
155 floor [173, 1; 474163, 1; 1398903353, 1; 281431596168986841220824577147, 1]
156 floor [3, 1; 31, 1; 139, 1; 1731115864310669, 1; 2886293162741841256597477513, 1]
157 floor [349, 1; 460451, 1; 1621163, 1; 495859674209478592267006001736547, 1]
158 ceil [3, 2; 127, 1; 106446763, 1; 2123467543911459232762462348605147931, 1]
159 floor [7, 1; 73816979890851693333312463143350322232292649851, 1]
160 ceil [5, 1; 13, 1; 199, 1; 6661, 1; 11994397616852450288191903404019096687369, 1]
161 floor [3, 1; 191, 1; 2411, 1; 1279111, 1; 66809843, 1; 17507082841337468780635214191, 1]
162 ceil [277, 1; 281, 1; 3833, 1; 46399, 1; 298614416565789429494073745279035121, 1]
163 ceil Mat([8267501747775389653330995872055236090016776783317, 1])
164 ceil [2753, 1; 20254212373, 1; 296539615604100858721655357946029957, 1]
165 floor [5, 1; 7602932747, 1; 808950198491, 1; 1075378495561112512933909589, 1]
166 ceil [3, 1; 7, 1; 11, 1; 28472714609, 1; 59033295413329, 1; 170343826963364790709541, 1]
167 ceil [37, 1; 179, 1; 1217, 1; 377593, 1; 49691507, 1; 874667203209056698133418313921, 1]
168 floor [3, 1; 31, 2; 409, 1; 928859, 1; 10902080041, 1; 22156296260948836369229555897, 1]
169 ceil [140249, 1; 2625241, 1; 133220759359, 1; 10787316296880324029486253053, 1]
170 ceil [5, 1; 11, 1; 2633, 1; 7307531842110623040612971526589581324601370219, 1]
171 floor [103, 1; 4027, 1; 2728027, 1; 1870452750940989251026804645421428625287, 1]
172 floor [3, 1; 29, 1; 33647, 1; 238451, 1; 2882141, 1; 26234701319677, 1; 80202546886692193793, 1]
173 floor [821, 1; 27271, 1; 166318024110673, 1; 167090473368877, 1; 13606268137367489, 1]
174 ceil [3, 3; 2803, 1; 24623, 1; 9086077162514538164993558952281075948464193, 1]
175 floor [73, 1; 463886125464219123562243275232030781160393393211829, 1]
176 floor [5, 1; 167, 1; 191537, 1; 724291, 1; 584671593454719673086112690797192351763, 1]
177 floor [2719, 1; 2939, 1; 23201, 1; 129497, 1; 5641817447069994136487658191015072323, 1]
178 ceil [21617, 1; 2953810697, 1; 660223757519, 1; 6426211292013553454275853353, 1]
179 floor [3, 1; 5, 1; 191, 1; 193, 1; 900917, 1; 25261216305351371, 1; 43055957437512795122463659, 1]
180 ceil [7, 1; 12461839571, 1; 456589317987028349, 1; 27206851038975809589627307, 1]
181 floor [309833622765786156559367, 1; 6994967036896282427471775689123, 1]
182 ceil [3, 1; 38923, 1; 266683, 1; 288733, 1; 1001180921147807, 1; 481517784182007064716859, 1]
183 ceil [5, 1; 239, 1; 26993, 1; 268754131132256262351333371491979595463861364719, 1]
184 ceil [3, 1; 7, 1; 11, 1; 195599, 1; 13310288676578377639, 1; 28829567002022948727810032119, 1]
185 ceil [347, 1; 5399, 1; 210678071, 1; 87856129092976918885059933596025745252339, 1]
186 floor [3, 4; 101, 1; 98297, 1; 203461, 1; 423873596546391121690024302668439270230369, 1]
187 ceil [1459, 1; 3041, 1; 7823, 1; 10357, 1; 25793, 1; 36998376517903, 1; 404325048152030443418957, 1]
188 ceil [11, 1; 73, 1; 847967, 1; 5797307745527671, 1; 70275401455149529011073099931743, 1]
189 floor [5, 1; 29, 1; 827, 1; 6113, 1; 1794326249, 1; 421817763454882503160233724272441986051, 1]
190 floor [3, 1; 499, 1; 704299, 1; 5686333, 1; 1156126729530714091, 1; 160091242417128440110879, 1]
191 ceil [167, 1; 1973, 1; 3473947, 1; 494025170116280083, 1; 3924626688339759168971162653, 1]
192 ceil [5, 1; 23, 1; 12886689710243, 1; 1193154236478401, 1; 2510200259412351752319743221, 1]
193 floor [274783, 1; 772903, 1; 41798366684808223712908004106422842268626755361, 1]
194 floor [3, 1; 53, 1; 467, 1; 691, 1; 49831, 1; 1257387668503, 1; 5522596448891133827080886819624261, 1]
195 floor [191, 1; 31481, 1; 3288091, 1; 1796008722921721910451277580107447075625536819, 1]
196 ceil [3, 2; 13, 1; 41, 1; 1483021, 1; 1838556830627807, 1; 5429627982322550170566340236195641, 1]
197 floor [11, 1; 719, 1; 1399, 1; 3547, 1; 243917, 1; 1222933109, 1; 922783403803, 1; 13147679882772669973559, 1]
198 floor [5, 2; 5147, 1; 12967, 1; 10020323, 1; 16990600750774904745162849168262500195513509, 1]
199 ceil [367, 1; 2287, 1; 2399, 1; 370663, 1; 445229, 1; 12462548735347, 1; 137189891311817644345262249, 1]
200 floor [3, 1; 7, 1; 163, 1; 839, 1; 107184613773481154938813, 1; 3691329620516885421892958999641, 1]
201 ceil [11, 1; 29, 1; 293, 1; 6659, 1; 166219, 1; 18380563, 1; 5770685530481, 1; 207099918973236794328020843, 1]
202 ceil [5, 1; 89, 1; 283, 1; 10461540925869331, 1; 3449864370960774710627701352753037457753, 1]
203 ceil [17, 1; 4457, 1; 66663392797760999, 1; 1799679432645902156754536611098921646439, 1]
204 floor [200220130997, 1; 45318716722464916587397, 1; 2003635753944911960563320713, 1]
205 ceil [3, 1; 5, 1; 359, 1; 119971, 1; 10639659926949841, 1; 5289862455495664663339367403453988841, 1]
206 floor [281, 1; 1522387, 1; 3303574733513, 1; 8607800442335354921, 1; 5978006685225713819599, 1]
207 floor [37, 1; 3930903482956732395488273129426416978991709363966675737822447, 1]
208 floor [47517829, 1; 86170728147527, 1; 2781680707143341, 1; 25538798924384067293712593, 1]
209 floor [11, 1; 17, 1; 5779, 1; 292279283428879, 1; 110614934240661691, 1; 16651294253466494072732147, 1]
210 ceil [13, 1; 14196691, 1; 351035161, 1; 139029641699998535357, 1; 129180091201992738720460309, 1]
211 floor [3, 1; 7, 1; 199, 1; 62099, 1; 2209849, 1; 4057836182669037122395007737790442606689508484253, 1]
212 ceil [5, 2; 1733, 1; 4271, 1; 42283, 1; 40179913, 1; 14804752619486162634544016750480336942422003, 1]
213 floor [467, 1; 2300189, 1; 4361207, 1; 1986950168102223487384693341878856021411061386389, 1]
214 floor [3, 2; 11, 1; 89, 1; 181, 1; 34412842066483, 1; 2739022623923441, 1; 123846613991463225703839015263, 1]
215 ceil [7, 1; 3557, 1; 5653, 1; 366139, 1; 3022349566151, 1; 239046557943954784893261649391131560253, 1]
216 ceil [739, 1; 100767301192330633964993025932143495602190720150182280348184623, 1]
217 floor [13, 1; 41, 1; 31957, 1; 79333, 1; 2210963, 1; 89706283481, 1; 555702861213990059816093384389518647, 1]
218 ceil [3, 1; 398990807, 1; 248851299410623057288067111326883422822049939915816005847, 1]
219 ceil [17, 1; 53, 1; 135009577, 1; 2066305181, 1; 2370119374271163703365704105879979766643788429, 1]
220 ceil [3, 2; 5, 1; 71, 1; 6563, 1; 320521, 1; 161986878092813, 1; 26829751769358071, 1; 40790384807455625040499, 1]
221 ceil [359501, 1; 6628479861241651155363003377579838120062548226879787721519189, 1]
222 floor [3, 1; 11, 1; 4099, 1; 198281, 1; 682374660679336214083015829, 1; 260404322856581425263424355719, 1]
223 ceil [5, 1; 7, 1; 43, 1; 3269267, 1; 15380691679369, 1; 25978567634116842691, 1; 4848373582855517302119907, 1]
224 floor [13211073647, 1; 1442998625104090198705803991491944112088170936595800246347, 1]
225 floor [3, 1; 53, 1; 89, 1; 17477, 1; 69224673341053669344379, 1; 2226994242783741671379533960247414043, 1]
226 ceil [1972098851640638958366652021, 1; 38666542689602845365597808038431422704859, 1]
227 ceil [257, 1; 307, 1; 1553, 1; 45869, 1; 323087, 1; 83987046203481419143039595925027085101363753012997, 1]
228 ceil [7, 1; 23, 1; 2467943, 1; 767649572818908910133183360960290963443251522427430536799711, 1]
229 ceil [5, 1; 37, 1; 113761, 1; 1005908431617373, 1; 907818940997560299781, 1; 31741773739805674349914121, 1]
230 ceil [491, 1; 1291, 1; 3517, 1; 3637, 1; 4074793, 1; 67249883, 1; 549115264568444606374477750498903574584339, 1]
231 floor [7, 1; 574384219762893721, 1; 1832734743980391751, 1; 331141679856170791548425242938361, 1]
232 floor [3, 1; 5, 1; 19, 1; 563, 1; 1223, 1; 240578044861929992273761079, 1; 103373279209613419126666985875245221, 1]
233 ceil [47, 1; 59685709684521751163349430084199, 1; 3479411317689287299550539940925277807, 1]
234 ceil [101, 1; 3769, 1; 109624922989, 1; 2139189994465003176740243, 1; 218674335966048300390961150807, 1]
235 floor [17, 1; 391417962993373, 1; 5867381259331386802719011449715473665344730338369488541, 1]
236 floor [3, 2; 35393802077711167, 1; 245128750477204554350422390101374118664959044214702821, 1]
237 ceil [107, 1; 307, 1; 8641, 1; 1555343, 1; 4024622590935763, 1; 87893442893175932006579825968123447888867, 1]
238 ceil [7, 1; 9137, 1; 30948142231, 1; 31140764569, 1; 5067088037833623169663824813457873246039990853, 1]
239 floor [5, 1; 170363, 1; 655604003, 1; 868252991979814525396225967, 1; 1288311243931794009960268929307, 1]
240 floor [3, 1; 43, 1; 7883, 1; 6008172910897276005729947857, 1; 204484456631720617613527717180374685289, 1]
241 floor [78043625687848246709, 1; 32016696554344534161292732810821374267675802120373547, 1]
242 ceil [3, 1; 11, 1; 5809289, 1; 26067959087369937053357642303825052686208812946835791028993903031, 1]
243 ceil [47, 1; 2017, 1; 255469, 1; 7117132505883300422032163147219, 1; 57986504913798765644459530120037, 1]
244 floor [3, 1; 5, 1; 7, 1; 13, 1; 2143, 1; 5602451, 1; 1219750640825124893243251245235191610456434069862462812677353, 1]
245 floor [857, 1; 498097051, 1; 1509737183, 1; 311938339481, 1; 92588745063827, 1; 2147884096405388560623623293, 1]
246 ceil [3, 1; 67, 1; 3307, 1; 4611460292353, 1; 49667973447624637805940193, 1; 525192807082924305108341462381, 1]
247 ceil [5, 1; 281, 1; 26575016869, 1; 2343999007661, 1; 12518367291785091923, 1; 145961710743038275093001865611, 1]
248 floor [3, 1; 5051, 1; 604319, 1; 23347815557321, 1; 1370377362093037, 1; 107130046860433169, 1; 10189698415161426859, 1]
249 ceil [11, 1; 89, 1; 1051, 1; 1855421, 1; 6596911, 1; 47928914430192167, 1; 1059711944401382426997038718061894756183, 1]
250 ceil [17653413313, 1; 68196802334259121, 1; 33890055926099146349, 1; 31355875525471958906904231401, 1]
251 ceil [17, 1; 137193733710317, 1; 1097060849822799151503342582559046543562085981472773713038077, 1]
252 floor [5, 1; 53, 1; 139, 1; 75571, 1; 1838349436051999465191785453769859418699779049691161054972806526233, 1]
253 ceil [3, 3; 49023122883188497193, 1; 7732308405369672266056440699769180825001915505059727601, 1]
254 floor [7, 1; 60127492477713677, 1; 8080817962574700619808377, 1; 6018350054450339204653502475958807, 1]
255 ceil [251, 1; 113826667, 1; 455268351604700188504067, 1; 3147376477536457121474028392039672027975537, 1]
256 floor [3, 1; 5, 1; 19, 1; 61, 1; 1493, 1; 3154491702625448435043767725292266154238230425573132371036916543361177, 1]
257 ceil [36677, 1; 20313321609211, 1; 3498534887098362913229, 1; 62825062456033283926373777961580635217, 1]
258 floor [239, 1; 751, 1; 2342687, 1; 3713317, 1; 3029429023, 1; 69238710392021308676744386381652419778847865139857, 1]
259 ceil [3, 1; 13, 1; 17, 2; 16079724421949983, 1; 34394726606170406191609, 1; 105080218265651184785424049972152059, 1]
260 ceil [5, 1; 7, 1; 2029, 1; 9787, 1; 135333301, 1; 31551449501531, 1; 441428332244079645509444624234389589920227889583, 1]
261 floor [3, 1; 277, 1; 124680737556814667, 1; 25287941838820280888331049551804447781047859216133310888677, 1]
262 ceil [19, 1; 23, 1; 233, 1; 701, 1; 83077, 1; 883705758222645487486993596399850247731797829182671194574489497727, 1]
263 floor [157109, 1; 855104197, 1; 29494545388973992549092631884527, 1; 2644917029248232479237761522143827, 1]
264 ceil [3, 2; 5, 1; 43, 1; 757, 1; 400570965959, 1; 39789085535828387789, 1; 897808231569621773008683445494044736812483, 1]
265 floor [22961, 1; 931000331, 1; 62616897471977, 1; 31318551945244391913631746714330023197149606608714167, 1]
266 floor [7, 1; 17, 1; 4951, 1; 142306245616526440936933936408757316469277254659619013816588411961890250331, 1]
267 floor [11, 1; 13, 1; 37, 1; 90864959017423139762594151218267, 1; 348786484683604333569620034576284576902975007, 1]
268 floor [45309196147, 1; 602135644503517, 1; 36785758657848930413, 1; 334166899266561516476165131292188957, 1]
269 floor [7, 1; 743, 1; 2334251856251548031, 1; 55248344514653678449503074341323941015525905775527719191249, 1]
270 floor [3467, 1; 289033889, 1; 1338693711663824586479357281016389362430718734221126128979386978617853, 1]
271 ceil [193, 1; 2221, 1; 253689381433021405122439, 1; 24672074628003493135913410408322294666976808724876237, 1]
272 ceil [3, 1; 37567, 1; 2001596071, 1; 12385492391960626752906125029, 1; 1920553176486233870339065692879129098123, 1]
273 floor [653, 1; 21576719, 1; 2719205817286618982989, 1; 280112926467400776480552010432538211133079920734431, 1]
274 ceil [199, 1; 67503010154647052685861885444979, 1; 1597819061949767746605962636518234626326416359287, 1]
275 ceil [3, 2; 79, 1; 619, 1; 1241128331, 1; 1547881386725080303513498611832481, 1; 50771406288153172747026285695348747, 1]
276 ceil [5, 1; 17, 1; 433, 1; 27226097, 1; 44661691189, 1; 9722252422709, 1; 14613178481141, 1; 590962456868393, 1; 22848882203814068081, 1]
277 ceil [3, 1; 11, 1; 19, 1; 23, 1; 103, 2; 1283, 1; 2341, 1; 25349, 1; 202357705591, 1; 27918828758198243, 1; 76185311999095661, 1; 34248920026313496311, 1]
278 ceil Mat([343418586823242820322805474836525626928691654043865394181347647845068459611197194017, 1])
279 floor [3, 1; 151, 1; 54581198989, 1; 658515335799386695367, 1; 42183889950003982632990968519121854512522562056247, 1]
280 floor [7, 1; 53, 1; 78842065075342159, 1; 46962573462384770009852746853677712743825668826463512764278267903, 1]
281 ceil [3, 1; 5, 1; 13, 1; 578374861299037420338895061, 1; 24359578203138517402332882741300125787929683085503475513, 1]
282 ceil [17, 1; 19, 1; 198091, 1; 211403, 1; 5672526012151, 1; 71612505820579401637085373228015883971239784446312969655761, 1]
283 ceil [3, 1; 89, 1; 246676399, 1; 200953668530135782458922363, 1; 830307526821652705310969061202647533341075643903, 1]
284 ceil [31, 1; 307, 1; 1173829, 1; 340227555563611679, 1; 5782682951262420822349495242920029455816133645275089143759, 1]
285 ceil [3, 2; 5, 1; 911, 1; 9791, 1; 4350009463, 1; 8836112437, 1; 315435597964831, 1; 9032618888408254611713431590581633949928121, 1]
286 floor [11, 1; 211, 1; 151057, 1; 5064512471, 1; 5864519853526140408145493099, 1; 8442628131301282673812709416857874978453, 1]
287 ceil [113, 1; 605509, 1; 190667017967, 1; 334485074177489, 1; 86115677584662136980241, 1; 467907677617166109757808160889, 1]
288 ceil [3, 1; 61, 1; 241, 1; 257, 1; 61651, 1; 463823, 1; 1409112589, 1; 131688507915331, 1; 5847046126305667820287109524869207188163343481, 1]
289 ceil [13, 1; 17, 1; 42451, 1; 246403, 1; 41237112981319, 1; 25939069377616137593965297, 1; 284436231269148358807362993902288707, 1]
290 ceil [3, 1; 5, 3; 29, 1; 67, 1; 186723023336659, 1; 933482236372677847, 1; 11075816393849518317205800481468800244530510549373, 1]
291 floor [547, 1; 37123, 1; 328122719, 1; 3290669111169971310555823963914121, 1; 128310687631156146446682923575104189871, 1]
292 floor [89, 1; 127, 1; 23754923, 1; 15444892057355726604339508836054700001, 1; 1356786469710396705747172881210418121071, 1]
293 floor [23, 1; 793517, 1; 850433441471, 1; 12259733818483, 1; 59138225621928516335296068932904706501698338055394521673, 1]
294 ceil [17, 1; 113, 1; 601, 1; 1910773281482028232393209157309, 1; 10202174103532890273433835035271732183409472898311091, 1]
295 floor [3, 1; 73, 1; 18919, 1; 343043107703, 1; 39556343318987631593, 1; 800620295867903856820955710102453211781059626908463, 1]
296 ceil [5, 1; 17372699, 1; 1036397649256366738302453964067945619107254951664966392294889789971007110366877101, 1]
297 floor [411719750392103350609, 1; 938814710428267341408277, 1; 465813560755517601925126694095997496409417873, 1]
298 ceil [3, 2; 101, 1; 16693, 1; 106782433, 1; 126113194441785913, 1; 33809884890679644710544575999, 1; 52122039724680270461046948077, 1]
299 floor [343823, 1; 2153267, 1; 189984292991, 1; 423156170461771, 1; 41863119730465168869170959, 1; 289048640269520737402926523, 1]
300 floor [5, 1; 293, 1; 336464053511316319436666519, 1; 2922182959971486166022022547665067239732484751087880579528949, 1]
301 floor [1459, 1; 2399, 1; 176887, 1; 226249221878907586016552846121197743, 1; 20565778147004092284813388211154964395895051, 1]
302 ceil [13, 1; 1657, 1; 94099, 1; 2312099, 1; 1229380720011809849009673718630391342651739368027586708340432571055578826043, 1]
303 ceil [3, 1; 5, 2; 37, 1; 71, 1; 173, 1; 1745928465986058950941098167, 1; 193633191296222569840028803549209075155641655397373508023, 1]
304 ceil [151, 1; 5651, 1; 3394465556052379, 1; 21100685465649560906938128259, 1; 377079966158561957212485278981309358641509, 1]
305 floor [3, 1; 3391, 1; 54722539, 1; 11197698894247101457460615377, 1; 7394173053288848137349195400095503434805545343031263, 1]
306 ceil [5, 1; 13, 1; 17, 1; 53, 1; 59, 1; 560091041, 1; 6235761439, 1; 1990841686159, 1; 2549001400527842647, 1; 1505284529344374992521220369425315091, 1]
307 floor [389, 1; 441073, 1; 3473136984201951799, 1; 309393854801316279140080954168495566699871374435761369473410759863, 1]
308 ceil [3, 2; 1139703727, 1; 2459802013, 1; 9236508498871, 1; 1582272215476954477026934558025689329172897077970390181903911, 1]
309 ceil [401, 1; 1291, 1; 36522131, 1; 27949996346345223324296389, 1; 1395549296354572633847104296021333349181037292017608753, 1]
310 ceil [3, 1; 149, 1; 1063, 1; 84948245670751, 1; 6356562108202237, 1; 99500766652061408261, 1; 57774970749769475730493428849263589119, 1]
311 floor [5, 2; 11239, 1; 5654219, 1; 6057625700501965950011, 1; 306528537621155817883364106074498388082767879511697439085991, 1]
312 ceil [1783, 1; 355721, 1; 20045681, 1; 45613189, 1; 518119631, 1; 142290794448158241004578743, 1; 137995428217099335695481964566105281, 1]
313 floor [3, 2; 7, 1; 59, 1; 2789, 1; 34679, 1; 383959307, 1; 13229646217, 1; 6173929966021171593426601, 1; 1046576250914041135502101858279485998177, 1]
314 floor [13, 1; 47, 1; 2957, 1; 17519, 1; 14659103, 1; 98984033, 1; 513841589214853067060728572694185419463593462405773710732675231701069, 1]
315 ceil [3, 1; 2111, 1; 95066123813719, 1; 88969862019046611761279289056309, 1; 881161360361182726563656281096246379339228449, 1]
316 ceil [11, 1; 37, 1; 8533211, 1; 27180455982460037017841487256650675035339358935093096904591390445140647144144610707969, 1]
317 ceil [3, 3; 5, 4; 17, 1; 193, 1; 49109, 1; 1276554569, 1; 1686424853389, 1; 3989617302680707, 1; 8084346288750547009179907346857245145216954417, 1]
318 ceil [23, 1; 251, 1; 5323, 1; 8513, 1; 413591137139, 1; 9035498819816437898161, 1; 19540020140283424257197, 1; 19766719618381145902711287049, 1]
319 ?
320 floor [5, 1; 1137200653, 1; 6778877178186673050097, 1; 39184900164768501884910073071073158990344429706993098422456978739, 1]
321 floor [3, 2; 316633, 1; 389646559, 1; 41569541765107, 1; 65443850316488531055856733287101414686173034176946941019987607674531, 1]
322 ceil [17, 1; 5741, 1; 61902350179871479355149513433816678719994206381877523688370655855320149706448245559922542139, 1]
323 floor [5, 1; 11, 2; 19, 1; 31, 1; 34646939630225558821, 1; 25912582312841435127823, 1; 37768293136449027985350628088192418492623723745359, 1]
324 ceil [3, 1; 821, 1; 9811585335777371937676860840601228409314290800247016937740821598882144784247226020157142257637, 1]
325 floor [13, 1; 503, 1; 2558791, 1; 8854464569569, 1; 38647066010557261, 1; 5690106667399303277189500141, 1; 1483502217589995242437234666081, 1]
326 ceil [47, 1; 2056675292086780177233881553225602176352433892851778954694097327493338093923482356395492883451909, 1]
327 ceil [3, 1; 5, 1; 8843031851804843597, 1; 1457475073376179372616669008969259953775779739818956935264433418099792237217479, 1]
328 ceil [256079, 1; 3072137, 1; 11939107, 1; 1951653850742445783622482576787, 1; 21092807960283404035642624027933173267835660945127, 1]
329 floor [3, 2; 8669, 1; 150291351827, 1; 508956619324146859, 1; 271867362744627531449, 1; 476617770755712443892371860360894785931409941, 1]
330 ceil [5, 1; 2011, 1; 326572718913183713, 1; 471000745285241510107047453238128041867770450224433974586135202208571756261477, 1]
331 floor [7, 1; 17, 1; 1871, 1; 4211, 1; 29723, 1; 4072632225049388731096303073794547195112727, 1; 27254595562884302830798791529817360981937211, 1]
332 floor [3, 1; 29, 1; 971, 1; 563779189, 1; 15873472487, 1; 45421847767, 1; 180160383879844194733808378088529377360306805672347642408528982187, 1]
333 ceil [31, 1; 1033, 1; 697495424991083, 1; 85881197463671610367, 1; 1076094323835854179443107, 1; 5994069134454114734198312785557193759, 1]
334 ceil [239, 1; 528719, 1; 148110056220727, 1; 393722489228666027, 1; 3358195432189532175510755245699393204051871666489865064270193, 1]
335 ceil [47, 1; 1297, 1; 811887239435953315916536125868549201459094952305405416193814827815411799605877383557819858386567, 1]
336 floor [5, 1; 17, 1; 37, 1; 23209883, 1; 1356032018843289137073606295778468489031131457309422798178207652967115071266038599070152843, 1]
337 floor [3, 2; 7, 1; 181, 1; 461, 1; 20123, 1; 52387, 1; 35723765939029186094731734545102021587290095146693893340565734511017181323307775766717, 1]
338 floor [23, 1; 8923, 1; 16883, 1; 2257820933, 1; 50611090139397460115953631573659857198830408341154655403275471919050427729422868733, 1]
339 floor [3, 1; 1117, 1; 236308369937457609955027756236685840748409282948780465607920062495014922791642754637069009835329097, 1]
340 floor [5, 1; 7, 1; 13, 1; 43, 1; 193, 1; 48121, 1; 304331, 1; 12387433, 1; 1540530396068509465699627347764311, 1; 1500769308290632472173392601933607969945343607, 1]
341 floor [3, 1; 19, 1; 2120868733, 1; 26201422991692244475801397312189159532930753900079976041645523566431538639666728138020092611, 1]
342 floor [59, 1; 7906036977966465051883377569, 1; 13581028679254123510636059368418993723679612923092026455329080215858833773, 1]
343 floor [3, 2; 7, 1; 19900139924449, 1; 11700478155456674927550604578510892061, 1; 863720495527758105115190219653252979620344810876181, 1]
344 ceil [5, 1; 11, 1; 17, 1; 590201, 1; 1148971, 1; 22831789, 1; 374025614234394222822050804230936043407, 1; 4679954215856611037168694026975350228016017, 1]
345 ceil [4133, 1; 530592951846991991, 1; 23110355398099076270141575240352614165968883499961561558884510145000261918266142223, 1]
346 ?
347 floor [3, 1; 5, 2; 37, 1; 463, 1; 677, 1; 21701, 1; 10739426830957939024148779171351090509704024903685372582838706746319456647959797056150276291, 1]
348 ceil [7, 3; 14797, 1; 79883245185846565874920391377961677521137821780243854135805194588424578855948653575053911330342581, 1]
349 floor [599, 1; 8839, 1; 271590116377, 1; 37013751014933779, 1; 15235140799026421353872917584942502432877803694671499442533949113411927, 1]
350 floor [3, 2; 13, 1; 54679, 1; 6801001, 1; 37273847219305199561701009938247075415968901627760588903061924163830088483889912151622956321, 1]
351 floor [27945178554819168569, 1; 60112523792489205392730956079378875869079, 1; 1930819291589981743697796000500636962169297257, 1]
352 floor [3, 1; 5, 1; 432466246402277622283904620488913853015610440177725898889702129347676846093541305433466608002764966050721, 1]
353 floor [17, 1; 431, 1; 14017538806144772935275354749, 1; 126320987756029791302285774874715607287139492757446481248251245413671227797, 1]
354 floor [3, 1; 83, 1; 104208734072837981273230029033473217594122997633186963587880031168114902673142483236979905542834931578487, 1]
355 ceil [310886951, 1; 17869263273793, 1; 88568571176344485847, 1; 105473821924913351385822861108602271240659980077847691534472809087, 1]
356 floor [3181, 1; 115386527, 1; 45677525489, 1; 6190733645473795169408964321945289616346846937404794097172189537801555548980461676671, 1]
357 floor [3, 1; 1276057, 1; 180951072038543363, 1; 4045355449156890986804156282355157697627, 1; 74077152093374476458746880165056005429830017, 1]
358 floor [5, 1; 13, 1; 24623, 1; 20760305879, 1; 26191848733, 1; 477055850408544440779198786617721050859743354052710294843657976962650993912145851, 1]
359 ceil [3, 4; 677, 1; 1296109, 1; 291104771, 1; 57117681487, 1; 82619250620719, 1; 8504270911587909271977710057225180648152396532240889928030098389, 1]
360 ceil [7, 1; 9448591, 1; 34400027843, 1; 1016259289467599, 1; 718215921712566779652034969867678496327942477652270121469380289669943120329, 1]
361 ?
362 ceil [37, 1; 41, 1; 44017, 1; 99480374623326890709777978462622763286950458377161580675702328450768554649441780987717307743648782287, 1]
363 ceil [5, 1; 29, 1; 238095176773, 1; 384817512737317184608716807188625542171405722975070913607249230563336098904735018355935908571281, 1]
364 ceil [7, 1; 1255307, 1; 3023816505098062763374457390313839805069779224697279996934534605834186426563061330339588915095200902381, 1]
365 ceil [83, 1; 137, 1; 4673419431704500415640330645121601816775851806264968644408284025524802642509397204437989340548743208892147, 1]
366 ceil [639490151, 1; 24766414418174413, 1; 68452452218837960777773, 1; 2184903088035650657182357159, 1; 44868880110377905188592044044445353, 1]
367 floor [5, 1; 7, 1; 13, 1; 11171, 1; 20269, 1; 221784245983, 1; 13242274454288352006984256688306587, 1; 702529169887331232907499395659315178361698319978049661, 1]
368 floor [3, 1; 29, 1; 869788617031, 1; 293889250949522203, 1; 19116427417510001572440809100162698685607245280819332968066406790897957644187801, 1]
369 floor [48557294033197, 1; 12317606882841205481, 1; 1421584193892342961426146587571537613300812582775097171884108612429748182900619, 1]
370 floor [3, 2; 7, 1; 257, 1; 19219, 1; 4500491, 1; 1099030789429, 1; 122902436589926339, 1; 8989772229487156978067149886306309810397861000753818902581984716863, 1]
371 ceil [5, 1; 37, 1; 1159393, 1; 103238908968913, 1; 296770082994436119126233150097613, 1; 517544224846484240569057659505384327189448049257167793123, 1]
372 floor [131, 1; 51924472532921525961523004351196564771720999952648753300095606868267511057562729147275116254172588974229108799, 1]
373 floor [3, 1; 167, 1; 95239, 1; 230311, 1; 2133187543927474687, 1; 580332793146445259044297195229655937111068678139444342091350332993998499982316993, 1]
374 ?
375 floor [43, 1; 156026651453863, 1; 372017576622197702819141, 1; 21802318169221694045608141231185485861509141664782948237083138532817158253, 1]
376 floor [3, 2; 5, 1; 17, 1; 5385939137, 1; 146056525680089149757, 1; 6795871050119574524495918254552810139227, 1; 26611809135492119250568039439190131884217, 1]
377 ceil [207864407, 1; 341377454180178752420313191, 1; 3067456421202576495954577256367774968209621584805529308604333945625491650939063, 1]
378 ceil [11, 2; 317, 1; 541, 1; 5011, 1; 182617661539, 1; 22925260830786595747367411071248677187152903734371015856521268855455985630561320368926939157, 1]
379 floor [7, 1; 1063553, 1; 11567943848685098921, 1; 7962644354199791387890896383007869363, 1; 1269646124232727555090758256922626272629119834751837, 1]
380 floor [3, 1; 347, 1; 571, 1; 1483, 1; 14451614112537179, 1; 136690713103098994352540354384632266437885821361286697239595308778477362153800000613241009, 1]
381 floor [41, 1; 84943371261661282665640754825450146155338314849367248325463720582157036625679204130878941399996680187304750862607, 1]
382 ?
383 ceil [269, 1; 186419, 1; 1200349, 1; 257513817649727, 1; 898716706869287718578004274860127319904139788689091478617711631858085221753805341852633, 1]
384 floor [3, 1; 61154273, 1; 77460182533192911437, 1; 41962370164525312882129, 1; 46721496856452318446776155876911316886564954345065080505620934031, 1]
385 ceil [5, 1; 1033, 1; 6511073549, 1; 6964412033, 1; 86262701827, 1; 26016958290699604746471583169440361, 1; 106009886940625532742757990020799069388746398817, 1]
386 ceil [13, 1; 241687637273, 1; 35470355408633153503669381873859310741236814901299925445088499786056478636379056473308052986853810672161, 1]
387 ceil [11, 1; 19, 1; 59, 1; 11287, 1; 4648964957893, 1; 13813458069737198743987, 1; 24937781472605891795521358532224704068664781363338055467482558195473258851, 1]
388 ?
389 floor [5, 1; 29509019, 1; 6042665293362661922841727558940029379129349929653429945767205688744314112394782941159144388597704012091697559, 1]
390 floor [3, 1; 7, 1; 13, 1; 12269164956907, 1; 2681374306393903661, 1; 21081546043303610615540131, 1; 9417719642962508637409624896492001314037592943630802630111, 1]
391 floor [2179, 1; 1636650986254973516032244814406470369979680500521264136620591484589849025099823637554300814932290326289032886744237, 1]
392 floor [3, 2; 31, 1; 177853519, 1; 76289208660757, 1; 1884141548538234948808943156148398410514810034420896310298144658766979928195253799851872434571, 1]
393 ceil [5, 1; 7, 1; 11, 1; 29, 2; 19853206385282168454163495003788641, 1; 2219146202288825120925575707530899933384132786999602282443189132934955215229447, 1]
394 floor [13, 1; 17, 1; 1873, 1; 20130449, 1; 63986518243, 1; 1230208392009953003, 1; 204231507971478852293475943163, 1; 212975610836671768580566520345717738065582993171, 1]
395 floor [3, 1; 19, 1; 110311, 1; 2206811, 1; 165242009353, 1; 230108249050583, 1; 108148827887082632442265560539389212020161554464107388235933217358636279723306593, 1]
396 floor [14821, 1; 155163637, 1; 274936841, 1; 4754893959299969, 1; 41344360116322879858974757, 1; 918133381741530173607181243707613927537485906488347952539, 1]
397 ceil [3, 1; 53, 1; 643, 1; 2232467697009630433715706963602890655201994619175967716546467580963467097472744751887991377360422153848052762246587, 1]
398 floor [13, 1; 28264361, 1; 6045515246254061, 1; 604875791032997325773, 1; 339735733829280118413185769900914518681682835585181948128276215445630368353, 1]
399 ceil [5, 2; 1376597581, 1; 26528107047623653977848183690843633050479559008400542750808301015018213738304804439289227482342638780849447999, 1]
400 ceil [3, 3; 11, 1; 97, 1; 617, 1; 12473, 1; 8743037, 1; 10939406293, 1; 86107886368893577166671690001661918218705058146792369921900558917821247354167173156733588781, 1]
401 ?
402 ceil [5, 1; 1669, 1; 159361, 1; 104879009447, 1; 52365625423124665262743486410273124681831070872020883172896125018413027460659037554451573279973737533, 1]
403 floor [3, 1; 7, 1; 67, 1; 9967, 1; 172848461055571385947069227413, 1; 22298203382829886776707682524346671, 1; 270258467339719324447994782109296165226425377141647, 1]
404 ceil [53, 1; 761, 1; 15628271, 1; 227684593, 1; 2880943501, 1; 3488948927, 1; 1104056894524069663, 1; 4265834048103641311, 1; 4300048788537611458775138350744425099856699411, 1]
405 floor [11, 1; 76198795689737956812810949, 1; 2765674941388291247601552972731, 1; 25205267954172474226149603915528649253583441323209350411710400073, 1]
406 ceil [3, 1; 5, 1; 29, 1; 78809, 1; 107558606449061, 1; 31692244749434963282069586729734845873429669057787412317839677053387321230130508667930268145518560061, 1]
407 ceil [162091, 1; 923591, 1; 17192980939, 1; 436992525377, 1; 308859121590053256091, 1; 672773724202510571989044460629586618090143129006325515494608608084833, 1]
408 ceil [3, 1; 13, 1; 17, 1; 5101, 1; 3348673, 1; 41274482868085616735436576275555680532871292510373140963188924527227992006260377515856531504243703325712236443, 1]
409 ceil [13022083328098679049233505977873, 1; 71791455560233587406204753710457081268990217737728113200104671140050488903797254880893366081, 1]
410 floor [3, 4; 5, 2; 37, 1; 7066087619, 1; 440142469683899639500035090773143, 1; 8023875444002884697629707164559263624412206070345731871767980498537922121793, 1]
411 ceil [29, 1; 41, 1; 67, 1; 633883, 1; 489993065190763, 1; 157398476311309039, 1; 960189606460405225893036337641455534433949746397153123222818778308899296037545867, 1]
412 floor [7, 1; 3509855333, 1; 610358890495175308859, 1; 7063418981691964121033, 1; 70608293661360965617870737150135493469816348734177894806311018022482093, 1]
413 floor [3, 1; 11, 1; 23, 1; 457, 1; 85889, 1; 2750823006154333529504851783, 1; 4542285597909833640977698593215416345691, 1; 40182837779088596142827005822861015979108253193, 1]
414 floor [16616732067989, 1; 1800352681094163516310709817554756292870868149506942292081586855124245169828227231561667939959363038109573005563, 1]
415 floor [3, 1; 5, 1; 7, 1; 857, 1; 9613, 1; 938059, 1; 73735050981822654682543251587405553566282040658120730977002850751275123559252754040617415081713656238283593377, 1]
416 ?
417 ceil [17, 1; 73, 1; 13191407, 1; 68125411, 1; 241139368185997205049379, 1; 889924244322441314844703465630921463230608348285566417102857403759894807952352174387, 1]
418 ?
419 ?
420 ceil [5, 1; 13, 1; 331, 1; 1549, 1; 4481, 1; 24169451, 1; 530455165986452302379562149965818511738634861899245890715396900313253409093593953377642656433820540568679841, 1]
421 ?
422 ?
423 floor [3, 1; 5680009, 1; 15907135071985061, 1; 3800341895604737088811, 1; 13704833667037010954717161, 1; 1084961886728328322371735670620916506552934106765858030429, 1]
424 ?