## Monday, July 16, 2018

### [xjzqtjzg] Top-heavy perfect powers

Below are a list of perfect powers b^n (b<=n) in increasing order, up to a googol.  There are 2413 values.  The list omits bases which are perfect powers themselves, e.g., 4^n, 8^n, 9^n.

Here is Haskell source code to compute the list.  We used functions from the data-ordlist package to merge an infinite list of ordered infinite lists into a single ordered list.  This seems to be a nontrivial function to write, having written it a few times ((1) and (2)) before discovering Data.List.Ordered.  (Incidentally, data-ordlist is not indexed by Hoogle version 4.  It is indexed by the alpha version (Hoogle 5), but Hoogle 5 cannot search by type Ord a => [[a]] -> [a].)  We also used Data.List.Ordered.minus to compute the set difference of two infinite lists.

The motivation was to investigate increasing the breadth of the Cunningham project, which currently factors integers b^n +/- 1 with b restricted to {2, 3, 5, 6, 7, 10, 11, 12}.  Within the original Cunningham range up to about 2^1200, there are 4357 perfect powers in the restricted set, but there would be 19256 in our expanded set.

What is the growth rate of this sequence?

2^2 2^3 2^4 3^3 2^5 2^6 3^4 2^7 3^5 2^8 2^9 3^6 2^10 2^11 3^7 5^5 2^12 3^8 2^13 5^6 2^14 3^9 2^15 6^6 3^10 2^16 5^7 2^17 3^11 2^18 6^7 5^8 2^19 3^12 7^7 2^20 3^13 6^8 5^9 2^21 2^22 3^14 7^8 2^23 5^10 6^9 3^15 2^24 2^25 7^9 3^16 5^11 6^10 2^26 3^17 2^27 5^12 2^28 7^10 6^11 3^18 2^29 2^30 3^19 5^13 7^11 2^31 6^12 3^20 2^32 5^14 2^33 10^10 3^21 6^13 7^12 2^34 5^15 3^22 2^35 2^36 6^14 3^23 7^13 10^11 2^37 5^16 2^38 3^24 11^11 6^15 2^39 7^14 5^17 3^25 10^12 2^40 2^41 3^26 6^16 11^12 5^18 2^42 7^15 3^27 2^43 12^12 10^13 6^17 2^44 5^19 3^28 7^16 11^13 2^45 3^29 2^46 5^20 10^14 6^18 12^13 2^47 3^30 7^17 2^48 13^13 11^14 5^21 2^49 6^19 3^31 10^15 2^50 12^14 7^18 3^32 2^51 5^22 6^20 13^14 11^15 2^52 3^33 2^53 10^16 14^14 7^19 5^23 12^15 3^34 2^54 6^21 2^55 11^16 3^35 13^15 5^24 2^56 7^20 10^17 6^22 2^57 3^36 14^15 12^16 2^58 5^25 15^15 3^37 11^17 7^21 2^59 13^16 6^23 10^18 2^60 3^38 5^26 14^16 12^17 2^61 7^22 3^39 2^62 6^24 11^18 15^16 5^27 13^17 2^63 10^19 3^40 2^64 12^18 7^23 6^25 14^17 3^41 2^65 5^28 11^19 2^66 15^17 10^20 3^42 13^18 2^67 6^26 5^29 7^24 2^68 12^19 3^43 14^18 2^69 11^20 17^17 5^30 3^44 10^21 6^27 2^70 7^25 13^19 15^18 2^71 3^45 12^20 5^31 2^72 14^19 6^28 11^21 3^46 7^26 2^73 10^22 17^18 2^74 13^20 15^19 5^32 3^47 6^29 2^75 18^18 12^21 7^27 2^76 3^48 11^22 14^20 10^23 5^33 2^77 6^30 17^19 3^49 13^21 2^78 15^20 7^28 12^22 5^34 2^79 18^19 3^50 11^23 10^24 14^21 2^80 6^31 19^19 3^51 2^81 5^35 13^22 7^29 17^20 2^82 15^21 3^52 12^23 6^32 2^83 11^24 10^25 18^20 5^36 14^22 2^84 3^53 7^30 19^20 2^85 13^23 6^33 3^54 17^21 5^37 15^22 2^86 12^24 10^26 20^20 11^25 2^87 7^31 3^55 18^21 14^23 6^34 2^88 5^38 3^56 13^24 2^89 19^21 12^25 10^27 7^32 15^23 17^22 11^26 2^90 3^57 6^35 5^39 20^21 2^91 14^24 18^22 3^58 2^92 21^21 13^25 7^33 5^40 2^93 10^28 6^36 12^26 11^27 19^22 3^59 15^24 2^94 17^23 2^95 20^22 3^60 14^25 5^41 7^34 6^37 18^23 2^96 13^26 10^29 21^22 3^61 12^27 11^28 2^97 5^42 15^25 19^23 2^98 17^24 22^22 6^38 7^35 3^62 14^26 2^99 20^23 10^30 5^43 3^63 13^27 2^100 18^24 11^29 12^28 6^39 2^101 21^23 7^36 3^64 15^26 19^24 2^102 5^44 17^25 22^23 14^27 10^31 2^103 3^65 6^40 13^28 20^24 11^30 7^37 12^29 2^104 23^23 18^25 5^45 3^66 2^105 21^24 15^27 6^41 2^106 3^67 19^25 17^26 10^32 14^28 7^38 5^46 2^107 22^24 11^31 13^29 12^30 3^68 2^108 20^25 18^26 23^24 6^42 2^109 5^47 3^69 15^28 7^39 10^33 21^25 2^110 24^24 17^27 14^29 19^26 11^32 3^70 2^111 13^30 12^31 6^43 5^48 22^25 2^112 7^40 20^26 3^71 18^27 10^34 2^113 23^25 15^29 6^44 5^49 2^114 3^72 11^33 21^26 14^30 17^28 24^25 19^27 13^31 12^32 2^115 7^41 3^73 22^26 2^116 5^50 10^35 6^45 20^27 18^28 2^117 15^30 3^74 23^26 11^34 7^42 2^118 14^31 12^33 13^32 5^51 17^29 21^27 3^75 6^46 19^28 2^119 24^26 10^36 2^120 22^27 3^76 7^43 5^52 18^29 2^121 20^28 11^35 15^31 6^47 14^32 12^34 2^122 3^77 13^33 23^27 26^26 17^30 10^37 21^28 2^123 5^53 19^29 7^44 3^78 24^27 2^124 6^48 11^36 22^28 2^125 15^32 18^30 3^79 20^29 5^54 12^35 14^33 13^34 2^126 10^38 7^45 23^28 6^49 17^31 3^80 26^27 2^127 21^29 19^30 5^55 11^37 2^128 24^28 3^81 15^33 2^129 12^36 7^46 6^50 18^31 22^29 14^34 13^35 10^39 20^30 3^82 2^130 5^56 17^32 2^131 23^29 11^38 3^83 26^28 19^31 21^30 6^51 7^47 2^132 5^57 12^37 15^34 10^40 24^29 2^133 3^84 13^36 14^35 18^32 22^30 20^31 2^134 6^52 28^28 5^58 3^85 7^48 17^33 11^39 2^135 23^30 19^32 2^136 21^31 10^41 12^38 3^86 26^29 15^35 13^37 5^59 2^137 6^53 14^36 24^30 7^49 18^33 3^87 2^138 22^31 20^32 11^40 17^34 2^139 5^60 28^29 3^88 10^42 6^54 12^39 2^140 19^33 23^31 7^50 21^32 13^38 15^36 14^37 29^29 2^141 26^30 3^89 5^61 18^34 11^41 2^142 24^31 6^55 20^33 3^90 22^32 10^43 2^143 17^35 7^51 12^40 5^62 2^144 28^30 3^91 13^39 19^34 15^37 14^38 23^32 6^56 21^33 2^145 11^42 26^31 29^30 3^92 18^35 7^52 2^146 10^44 5^63 24^32 20^34 12^41 2^147 17^36 22^33 30^30 6^57 3^93 2^148 13^40 15^38 14^39 5^64 19^35 11^43 7^53 3^94 2^149 28^31 23^33 21^34 10^45 6^58 2^150 18^36 26^32 12^42 3^95 29^31 5^65 2^151 17^37 20^35 24^33 7^54 22^34 13^41 2^152 30^31 3^96 11^44 14^40 15^39 6^59 10^46 19^36 2^153 5^66 31^31 21^35 3^97 23^34 28^32 2^154 12^43 18^37 7^55 2^155 6^60 26^33 17^38 3^98 13^42 29^32 5^67 20^36 11^45 24^34 2^156 22^35 14^41 10^47 15^40 3^99 2^157 30^32 19^37 7^56 6^61 12^44 5^68 2^158 21^36 23^35 18^38 3^100 31^32 28^33 2^159 13^43 11^46 17^39 10^48 26^34 14^42 20^37 2^160 7^57 3^101 15^41 5^69 6^62 29^33 24^35 22^36 2^161 12^45 19^38 3^102 30^33 2^162 21^37 5^70 11^47 18^39 10^49 13^44 7^58 23^36 6^63 2^163 3^103 28^34 31^33 17^40 14^43 2^164 15^42 20^38 26^35 3^104 5^71 12^46 22^37 2^165 24^36 29^34 6^64 7^59 19^39 2^166 11^48 10^50 3^105 33^33 13^45 18^40 30^34 21^38 2^167 5^72 23^37 14^44 17^41 15^43 2^168 3^106 6^65 28^35 7^60 31^34 12^47 20^39 2^169 26^36 10^51 22^38 5^73 11^49 3^107 24^37 19^40 2^170 29^35 13^46 6^66 18^41 2^171 3^108 7^61 21^39 14^45 33^34 17^42 30^35 5^74 23^38 15^44 2^172 12^48 10^52 3^109 20^40 11^50 34^34 2^173 28^36 6^67 31^35 26^37 22^39 13^47 2^174 7^62 5^75 19^41 24^38 3^110 29^36 2^175 18^42 14^46 12^49 21^40 17^43 6^68 15^45 3^111 2^176 10^53 23^39 11^51 5^76 33^35 30^36 7^63 2^177 20^41 3^112 13^48 28^37 2^178 34^35 31^36 6^69 22^40 19^42 26^38 5^77 24^39 14^47 2^179 3^113 12^50 18^43 10^54 35^35 7^64 15^46 29^37 17^44 11^52 2^180 21^41 3^114 23^40 6^70 2^181 5^78 13^49 20^42 30^37 33^36 2^182 3^115 7^65 19^43 28^38 10^55 14^48 12^51 22^41 2^183 34^36 31^37 26^39 11^53 24^40 5^79 18^44 6^71 15^47 3^116 17^45 2^184 21^42 29^38 35^36 2^185 13^50 7^66 3^117 23^41 5^80 20^43 2^186 10^56 6^72 12^52 30^38 14^49 33^37 11^54 19^44 2^187 3^118 22^42 28^39 15^48 18^45 24^41 2^188 26^40 17^46 5^81 7^67 34^37 31^38 3^119 6^73 13^51 21^43 2^189 10^57 29^39 35^37 23^42 2^190 12^53 20^44 3^120 11^55 14^50 5^82 7^68 2^191 19^45 6^74 30^39 15^49 33^38 22^43 3^121 18^46 2^192 17^47 28^40 13^52 24^42 10^58 26^41 5^83 37^37 2^193 31^39 21^44 34^38 3^122 12^54 7^69 11^56 6^75 2^194 14^51 29^40 20^45 23^43 35^38 3^123 2^195 5^84 15^50 19^46 18^47 10^59 2^196 13^53 17^48 22^44 30^40 6^76 7^70 3^124 33^39 2^197 28^41 24^43 12^55 11^57 5^85 26^42 21^45 37^38 14^52 2^198 3^125 31^40 34^39 20^46 2^199 23^44 6^77 29^41 15^51 10^60 7^71 38^38 19^47 5^86 3^126 13^54 2^200 35^39 18^48 17^49 11^58 22^45 12^56 2^201 30^41 3^127 6^78 24^44 33^40 14^53 28^42 2^202 5^87 21^46 26^43 7^72 10^61 3^128 2^203 31^41 20^47 15^52 37^39 34^40 13^55 23^45 19^48 2^204 29^42 11^59 6^79 18^49 5^88 12^57 17^50 3^129 38^39 7^73 2^205 22^46 35^40 14^54 10^62 2^206 3^130 30^42 39^39 24^45 21^47 5^89 28^43 6^80 26^44 33^41 2^207 15^53 13^56 20^48 11^60 3^131 7^74 12^58 2^208 31^42 23^46 19^49 37^40 17^51 18^50 34^41 29^43 5^90 2^209 3^132 10^63 6^81 14^55 22^47 38^40 2^210 35^41 7^75 3^133 21^48 24^46 13^57 15^54 30^43 2^211 11^61 5^91 39^40 12^59 26^45 28^44 20^49 33^42 6^82 2^212 3^134 19^50 17^52 10^64 23^47 18^51 40^40 2^213 31^43 14^56 7^76 37^41 5^92 34^42 29^44 3^135 2^214 22^48 11^62 6^83 13^58 15^55 2^215 12^60 38^41 21^49 35^42 24^47 3^136 30^44 10^65 5^93 2^216 20^50 7^77 26^46 28^45 17^53 19^51 39^41 18^52 33^43 2^217 14^57 23^48 6^84 3^137 11^63 31^44 2^218 40^41 5^94 13^59 22^49 29^45 12^61 3^138 34^43 15^56 37^42 7^78 2^219 10^66 21^50 41^41 6^85 2^220 24^48 3^139 38^42 20^51 35^43 5^95 17^54 30^45 14^58 19^52 26^47 2^221 18^53 28^46 11^64 23^49 7^79 3^140 33^44 39^42 2^222 13^60 12^62 6^86 10^67 15^57 5^96 31^45 22^50 2^223 29^46 3^141 40^42 34^44 2^224 37^43 21^51 7^80 14^59 24^49 20^52 17^55 11^65 6^87 2^225 41^42 3^142 19^53 18^54 5^97 26^48 38^43 35^44 30^46 13^61 12^63 10^68 28^47 2^226 23^50 42^42 15^58 3^143 33^45 2^227 39^43 7^81 22^51 6^88 5^98 31^46 2^228 3^144 11^66 29^47 21^52 14^60 40^43 17^56 34^45 2^229 20^53 10^69 37^44 24^50 18^55 19^54 13^62 12^64 3^145 5^99 2^230 6^89 7^82 26^49 41^43 15^59 30^47 23^51 28^48 35^45 38^44 2^231 3^146 11^67 42^43 22^52 2^232 33^46 5^100 14^61 10^70 39^44 6^90 21^53 31^47 17^57 3^147 2^233 7^83 12^65 13^63 29^48 43^43 20^54 18^56 19^55 24^51 2^234 34^46 40^44 15^60 37^45 5^101 3^148 2^235 26^50 23^52 6^91 11^68 30^48 28^49 41^44 7^84 10^71 35^46 2^236 14^62 38^45 3^149 22^53 12^66 13^64 5^102 2^237 17^58 33^47 21^54 42^44 18^57 20^55 3^150 31^48 6^92 39^45 19^56 2^238 29^49 15^61 24^52 7^85 11^69 43^44 2^239 34^47 5^103 10^72 3^151 40^45 37^46 26^51 23^53 14^63 2^240 12^67 44^44 28^50 6^93 30^49 13^65 22^54 3^152 2^241 35^47 41^45 17^59 38^46 7^86 5^104 21^55 18^58 2^242 20^56 33^48 19^57 11^70 15^62 3^153 10^73 42^45 31^49 29^50 6^94 2^243 24^53 39^46 14^64 12^68 5^105 2^244 3^154 43^45 34^48 13^66 7^87 23^54 26^52 40^46 37^47 2^245 28^51 17^60 22^55 30^50 6^95 11^71 3^155 44^45 10^74 21^56 2^246 18^59 5^106 15^63 35^48 20^57 19^58 41^46 38^47 2^247 7^88 45^45 33^49 3^156 12^69 14^65 24^54 31^50 29^51 13^67 2^248 42^46 6^96 39^47 5^107 23^55 3^157 2^249 11^72 26^53 10^75 34^49 17^61 43^46 22^56 7^89 28^52 2^250 15^64 37^48 40^47 18^60 30^51 21^57 3^158 19^59 20^58 6^97 5^108 12^70 2^251 44^46 14^66 35^49 13^68 41^47 38^48 2^252 3^159 24^55 33^50 10^76 11^73 29^52 45^46 7^90 31^51 2^253 5^109 23^56 6^98 17^62 42^47 3^160 39^48 26^54 15^65 2^254 46^46 22^57 18^61 34^50 12^71 21^58 28^53 19^60 20^59 2^255 43^47 14^67 30^52 3^161 37^49 13^69 5^110 40^48 7^91 10^77 6^99 11^74 2^256 35^50 44^47 24^56 3^162 2^257 38^49 41^48 33^51 29^53 17^63 31^52 5^111 23^57 15^66 2^258 45^47 12^72 7^92 3^163 6^100 26^55 18^62 22^58 42^48 14^68 39^49 2^259 13^70 10^78 19^61 21^59 20^60 11^75 34^51 28^54 46^47 3^164 2^260 5^112 30^53 43^48 37^50 40^49 2^261 47^47 6^101 7^93 24^57 3^165 35^51 17^64 12^73 15^67 2^262 44^48 33^52 29^54 23^58 5^113 38^50 10^79 41^49 31^53 18^63 14^69 13^71 11^76 2^263 3^166 22^59 26^56 19^62 21^60 45^48 20^61 6^102 7^94 2^264 42^49 39^50 28^55 34^52 3^167 5^114 30^54 2^265 46^48 12^74 15^68 37^51 17^65 10^80 43^49 24^58 2^266 40^50 6^103 3^168 11^77 13^72 14^70 47^48 7^95 35^52 18^64 23^59 2^267 5^115 29^55 33^53 44^49 31^54 22^60 19^63 38^51 3^169 41^50 26^57 21^61 20^62 2^268 48^48 6^104 12^75 2^269 10^81 45^49 28^56 5^116 3^170 7^96 39^51 15^69 42^50 34^53 17^66 11^78 30^55 2^270 13^73 14^71 24^59 46^49 37^52 2^271 3^171 18^65 43^50 23^60 40^51 6^105 5^117 35^53 19^64 2^272 22^61 29^56 47^49 20^63 7^97 21^62 33^54 10^82 12^76 31^55 3^172 26^58 38^52 44^50 2^273 41^51 11^79 15^70 48^49 13^74 17^67 5^118 2^274 6^106 28^57 14^72 3^173 45^50 34^54 30^56 39^52 2^275 42^51 24^60 7^98 18^66 10^83 3^174 23^61 2^276 12^77 37^53 19^65 46^50 5^119 22^62 6^107 20^64 21^63 43^51 40^52 11^80 29^57 35^54 2^277 26^59 3^175 15^71 31^56 33^55 13^75 47^50 7^99 14^73 17^68 2^278 38^53 44^51 41^52 5^120 28^58 3^176 2^279 10^84 6^108 48^50 18^67 12^78 24^61 30^57 34^55 2^280 45^51 39^53 11^81 19^66 42^52 23^62 3^177 7^100 20^65 22^63 5^121 2^281 21^64 13^76 15^72 37^54 46^51 14^74 6^109 29^58 2^282 26^60 17^69 40^53 35^55 3^178 43^52 50^50 10^85 31^57 33^56 2^283 12^79 5^122 47^51 38^54 7^101 18^68 28^59 11^82 3^179 44^52 41^53 2^284 24^62 6^110 30^58 19^67 48^51 34^56 13^77 23^63 2^285 15^73 20^66 3^180 22^64 39^54 21^65 14^75 45^52 5^123 10^86 42^53 2^286 17^70 7^102 37^55 29^59 26^61 12^80 3^181 6^111 2^287 11^83 46^52 35^56 31^58 40^54 33^57 43^53 18^69 50^51 5^124 2^288 28^60 3^182 13^78 38^55 47^52 24^63 19^68 2^289 10^87 15^74 7^103 51^51 41^54 44^53 14^76 30^59 23^64 6^112 20^67 22^65 21^66 34^57 2^290 3^183 17^71 5^125 12^81 48^52 11^84 39^55 2^291 45^53 42^54 26^62 29^60 3^184 37^56 18^70 7^104 2^292 6^113 31^59 10^88 13^79 35^57 5^126 33^58 40^55 46^53 2^293 15^75 43^54 19^69 14^77 3^185 28^61 24^64 50^52 38^56 20^68 12^82 2^294 23^65 11^85 21^67 17^72 22^66 47^53 30^60 41^55 6^114 7^105 3^186 44^54 5^127 51^52 2^295 34^58 10^89 39^56 2^296 48^53 13^80 18^71 26^63 29^61 3^187 52^52 45^54 42^55 15^76 37^57 14^78 2^297 5^128 31^60 6^115 19^70 35^58 11^86 12^83 7^106 33^59 3^188 2^298 24^65 40^56 28^62 20^69 46^54 17^73 43^55 23^66 21^68 22^67 10^90 2^299 50^53 38^57 30^61 5^129 3^189 13^81 6^116 47^54 2^300 41^56 34^59 18^72 44^55 7^107 51^53 14^79 26^64 15^77 11^87 2^301 12^84 3^190 29^62 39^57 48^54 19^71 5^130 42^56 2^302 45^55 52^53 37^58 31^61 10^91 6^117 17^74 20^70 24^66 35^59 33^60 3^191 28^63 2^303 21^69 23^67 7^108 22^68 40^57 13^82 53^53 46^55 43^56 2^304 5^131 30^62 3^192 38^58 18^73 11^88 14^80 12^85 15^78 50^54 2^305 6^118 34^60 41^57 47^55 26^65 10^92 44^56 19^72 3^193 2^306 7^109 29^63 51^54 5^132 39^58 17^75 20^71 2^307 13^83 31^62 48^55 24^67 37^59 42^57 21^70 3^194 45^56 23^68 6^119 28^64 22^69 33^61 35^60 52^54 11^89 2^308 12^86 14^81 18^74 15^79 40^58 7^110 5^133 10^93 2^309 3^195 30^63 43^57 53^54 46^56 38^59 2^310 19^73 6^120 26^66 34^61 50^55 17^76 3^196 41^58 54^54 13^84 29^64 2^311 47^56 5^134 20^72 44^57 11^90 7^111 24^68 39^59 21^71 12^87 51^55 2^312 31^63 23^69 22^70 14^82 3^197 10^94 28^65 15^80 37^60 18^75 33^62 42^58 48^56 6^121 35^61 2^313 45^57 5^135 52^55 3^198 40^59 2^314 30^64 19^74 7^112 13^85 43^58 17^77 11^91 46^57 38^60 26^67 2^315 53^55 6^122 3^199 34^62 12^88 20^73 10^95 29^65 5^136 2^316 14^83 50^56 41^59 21^72 24^69 15^81 54^55 47^57 22^71 44^58 23^70 18^76 3^200 2^317 31^64 39^60 7^113 28^66 51^56 37^61 33^63 6^123 55^55 2^318 35^62 5^137 42^59 13^86 11^92 48^57 45^58 3^201 19^75 17^78 10^96 30^65 2^319 12^89 52^56 40^60 26^68 14^84 20^74 2^320 7^114 38^61 43^59 3^202 15^82 46^58 5^138 34^63 6^124 29^66 21^73 53^56 24^70 2^321 22^72 18^77 23^71 41^60 50^57 11^93 3^203 13^87 2^322 31^65 28^67 44^59 47^58 10^97 54^56 39^61 12^90 5^139 33^64 19^76 7^115 17^79 37^62 2^323 6^125 35^63 51^57 3^204 42^60 14^85 55^56 30^66 48^58 2^324 45^59 20^75 15^83 26^69 40^61 3^205 52^57 2^325 21^74 5^140 11^94 56^56 18^78 38^62 29^67 24^71 22^73 10^98 43^60 34^64 13^88 7^116 23^72 6^126 46^59 2^326 12^91 53^57 3^206 41^61 28^68 31^66 17^80 2^327 19^77 50^58 5^141 14^86 44^60 39^62 47^59 33^65 2^328 54^57 3^207 15^84 37^63 35^64 6^127 7^117 20^76 11^95 30^67 10^99 42^61 2^329 51^58 26^70 13^89 21^75 18^79 45^60 48^59 55^57 3^208 5^142 12^92 40^62 22^74 2^330 24^72 23^73 29^68 38^63 52^58 34^65 6^128 2^331 43^61 56^57 17^81 14^87 3^209 7^118 19^78 46^60 28^69 31^67 2^332 5^143 15^85 11^96 41^62 10^100