A385412 Numbers k that are not perfect powers and whose trailing digits form a power of b, where 1 < b < k.
11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92
Offset: 1
Examples
Numbers having a single digit are not terms: 1 is not a term since 1 is a power of all bases b; Numbers k=2..9 are not in the sequence since k^e mod 10 <> b, with e > 1. 10 is not a term since it ends in a single zero, and zero is not a power of another number. a(1) = 11 since it is a prime and 11 mod 10 is a power of all bases b. a(2) = 12 since it is not a perfect power, and 12 mod 10 is a power of 2. a(3) = 13 since it is a prime and 13 mod 10 is a power of 3. All primes p greater than 7 are terms since they are not perfect powers. 20 is not a term because it ends with a zero, and zero is not a power of another number. 26 is a term since it is not a perfect power, and 26 mod 10 = 6^1.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
Programs
-
Mathematica
nn = 130; t = Union@ Flatten@ Table[m = 10^IntegerLength[b] + b; If[m > nn, Nothing, s = b^Range[0, Floor@ Log[b, nn]]; Flatten@ Reap[Map[(w = IntegerDigits[#]; i = 0; While[Set[k, FromDigits@ Join[IntegerDigits[i], w]] <= nn, If[FreeQ[s, k], Sow[k]]; i++]) &, s] ][[-1]]], {b, 2, nn}]; Select[t, GCD @@ FactorInteger[#][[;; , -1]] === 1 &]