factorint nextprime

The penultimate factor of p-1 of the next prime after the given power of two. See also factorint precprime

gettime();for(i=3,1000,x=factorint(nextprime(2^i)-1);y=matsize(x);if(y[1]>1,print(i," ",x[y[1]-1,1]," ",gettime())))

3 => 2 , 5 => 2 , 6 => 3 , 7 => 5 , 9 => 5 , 10 => 5 , 11 => 3 , 12 => 3 , 13 => 3 , 14 => 5 , 15 => 29 , 17 => 19 , 18 => 3 , 19 => 41 , 20 => 101 , 21 => 3 , 22 => 3 , 23 => 17 , 24 => 103 , 25 => 11 , 26 => 3 , 27 => 3 , 28 => 19 , 29 => 23 , 30 => 59 , 31 => 149 , 32 => 131 , 33 => 59 , 34 => 1277 , 35 => 7 , 36 => 883 , 37 => 953 , 38 => 35573 , 39 => 383 , 40 => 5 , 41 => 277 , 42 => 227 , 43 => 227 , 44 => 97 , 45 => 120739 , 46 => 1947973 , 47 => 331 , 48 => 1361 , 49 => 7 , 50 => 11 , 51 => 3517 , 52 => 612229 , 53 => 6529 , 54 => 7724869 , 55 => 246241 , 56 => 34501 , 57 => 246241 , 58 => 33829 , 59 => 1361 , 60 => 2971 , 61 => 2447 , 62 => 15467 , 63 => 7005787 , 64 => 7 , 65 => 1499567 , 66 => 5419 , 67 => 312709 , 68 => 5419 , 69 => 293 , 70 => 2 , 71 => 4787 , 72 => 60594931 , 73 => 60594931 , 74 => 1559689 , 75 => 7416361 , 76 => 971 , 77 => 10362841 , 78 => 389 , 79 => 3677 , 80 => 31039 , 81 => 78233 , 82 => 3 , 83 => 52825813 , 84 => 155377 , 85 => 14009 , 86 => 2846950409 , 87 => 126265247 , 88 => 676348286641 , 89 => 13614755069 , 90 => 1892113 , 91 => 35320663991 , 92 => 296131229 , 93 => 3491 , 94 => 3033169 , 95 => 17 , 96 => 619 , 97 => 4153421 , 98 => 9917839 , 99 => 1474408489 , 100 => 52203989 , 101 => 956903 , 102 => 15183787 , 103 => 23561 , 104 => 3544157 , 105 => 733 , 106 => 17 , 107 => 29207603 , 108 => 415141630193 , 109 => 3019 , 110 => 5 , 111 => 3121 , 112 => 28976429 , 113 => 379 , 114 => 1490834899 , 115 => 27726931 , 116 => 25781083 , 117 => 190570805678557 , 118 => 28343271991 , 119 => 59393 , 120 => 322747 , 121 => 28124221 , 122 => 7471602694511 , 123 => 2 , 124 => 187385987 , 125 => 379 , 126 => 198685121119 , 127 => 54721235939 , 128 => 5816689 , 129 => 229668251 , 130 => 66361823149 , 131 => 419 , 132 => 33773 , 133 => 11197 , 134 => 1132854612138368291 , 135 => 34110701 , 136 => 94841 , 137 => 1069 , 138 => 1883157135247 , 139 => 32081978728531 , 140 => 577 , 141 => 4506937 , 142 => 14155556315601773 , 143 => 111349165273 , 144 => 7 , 145 => 29 , 146 => 404011 , 147 => 35354098335041 , 148 => 84900373 , 149 => 29 , 150 => 66611352953 , 151 => 853 , 152 => 4078967803638757 , 153 => 28190690609748977 , 154 => 57638029 , 155 => 3183398673950160499 , 156 => 254572484479 , 157 => 140135408598613 , 158 => 600696432006490087537 , 159 => 39591902879 , 160 => 173848897619 , 161 => 2092970810101 , 162 => 738622015238517469867 , 163 => 72680077859 , 164 => 371631372010427 , 165 => 3033872055694266793 , 166 => 51581 , 167 => 19 , 168 => 213856863324036931 , 169 => 7673 , 170 => 349682762843012539 , 171 => 8562191377 , 172 => 996637 , 173 => 1184811007753 , 174 => 519812308091528579 , 175 => 64082150767423457 , 176 => 3541 , 177 => 2262418866139 , 178 => 203889220265798047 , 179 => 191177683417957 , 180 => 2620759002413 , 181 => 238166046827263 , 182 => 116646869443567827421 , 183 => 15459263 , 184 => 402091141946990581 , 185 => 422130953307364455748703 , 186 => 730933508801191 , 187 => 457507 , 188 => 42407 , 189 => 228226151 , 190 => 768614336404564651 , 191 => 127391413339 , 192 => 249964755824311260181 , 193 => 43691 , 194 => 14582476464855624168947 , 195 => 530973811979278664611 , 196 => 12983 , 197 => 78739038977031613 , 198 => 341117591230040819 , 199 => 27806552462203 , 200 => 62620177 , 201 => 135096265434489839 , 202 => 675889 , 203 => 22804045411650469 , 204 => 52626249439 , 205 => 575944527944101109 , 206 => 6713103182899 , 207 => 638624977 , 208 => 8572547189567 , 209 => 1248251259919 , 210 => 8831418697 , 211 => 5699217237190662496813 , 212 => 32322044425553 , 213 => 3185257 , 214 => 48831491 , 215 => 598101348759401 , 216 => 3 , 217 => 146919792181 , 218 => 329824456676414692409 , 219 => 84229 , 220 => 457 , 221 => 10269694647880783 , 222 => 562749712093113293639265552817 , 223 => 3354409408717 , 224 => 9907 , 225 => 7 , 226 => 21337199679782081 , 227 => 964143547970507 , 228 => 3454631579714210387 , 229 => 84379598431955155155523 , 230 => 891695404608302473 , 231 => 3313389311 , 232 => 58879676392546531901 , 233 => 7 , 234 => 30766081371962430877027 , 235 => 2740834682972131 , 236 => 22735307249651 , 237 => 2687861391381743 , 238 => 3651704087219939111 , 239 => 618399785157681981868109 , 240 => 6999441151774987463 , 241 => 8379052851628702138261 , 242 => 414745175048619315247 , 243 => 120562177 , 244 => 213973 , 245 => 307817541673849 , 246 => 2361846005176909259 , 247 => 64731702760358791 , 248 => 89494061 , 249 => 7 , 250 => 548379867989362859525425493551 , 251 => 15162868758218274451 , 252 => 1793551536392300393836110149 , 253 => 3271389001471 , 254 => 2 , 255 => 36139168099759229 , 256 => 1200461269659761894123 , 257 => 5977729210235941 , 258 => 92688479535572441677859 , 259 => 11927816443 , 260 => 151995302154206813139306866773 , 261 => 24187325339