A385411 Numbers k that are not perfect powers, not divisible by some b, and whose trailing digits form a power of b, where 1 < b < k.
11, 13, 14, 17, 18, 19, 21, 23, 26, 28, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 54, 56, 57, 58, 59, 61, 67, 68, 69, 71, 73, 74, 76, 78, 79, 83, 86, 87, 89, 91, 94, 97, 98, 101, 103, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124
Offset: 1
Examples
Numbers having a single digit are not terms: k=1..2 are not terms since b < k could not be a valid base; Numbers k=3..9 are not in the sequence since there is not b < k with the same digit of k. 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 = 1 = b^0 for all bases b in [10] \ 1. 12 is not a term since for all 1 < b < 12 either b | 12 or 12 mod 10 = 2 <> b^e, with e > 0. a(2) = 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 and are not divisible by 1 < b < p. 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, 6 does not divide 26, and 26 mod 10 = 6^1. 116 is a term since it is not a perfect power (116 = 2^2*29), 16 does not divide 116, and 116 mod 100 = 16^1.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
nn = 125; 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[And[!Divisible[k, b], FreeQ[s, k]], Sow[k]]; i++]) &, s] ][[-1]]], {b, 2, nn}]; Select[t, GCD @@ FactorInteger[#][[;; , -1]] === 1 &]