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A385410 Multiples k of b that are not perfect powers and whose trailing digits form a power of b, where 1 < b < k.

%I A385410 #20 Jul 05 2025 05:15:02
%S A385410 12,14,15,18,21,22,24,28,33,34,35,38,39,42,44,45,48,51,52,54,55,58,62,
%T A385410 63,65,66,68,69,72,74,75,77,78,82,84,85,88,91,92,93,94,95,96,98,99,
%U A385410 102,104,105,108,110,111,112,114,115,116,118,120,122,123,124,126,129
%N A385410 Multiples k of b that are not perfect powers and whose trailing digits form a power of b, where 1 < b < k.
%H A385410 Michael De Vlieger, <a href="/A385410/b385410.txt">Table of n, a(n) for n = 1..10000</a>
%e A385410 In general, no prime p is a term since they are a power of base p.
%e A385410 Numbers having a single digit are not terms:
%e A385410   1 is not a term since 1 is a power of all bases b;
%e A385410   Composites 4, 6, and 9 are not in the sequence since 4 = 2^2, 6 = 2*3, and 9 = 3^2.
%e A385410 10 is not a term since it ends in a single zero, and zero is not a power of another number.
%e A385410 a(1) = 12 since it is not a perfect power, 2 | 12, and 12 mod 10 is a power of 2.
%e A385410 a(2) = 14 since it is not a perfect power, 2 | 14, and 14 mod 10 is a power of 2.
%e A385410 20 is not a term since it ends with a zero, and zero is not a power of another number.
%e A385410 26 is not a term since 6 does not divide 26.
%e A385410 1100 is a term since it is not a perfect power, 100 = 100^1, and 100 | 1100.
%t A385410 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[And[Divisible[k, b], FreeQ[s, k]], Sow[k]]; i++]) &, s] ][[-1]]], {b, 2, nn}]; Select[t, GCD @@ FactorInteger[#][[;; , -1]] === 1 &]
%Y A385410 Subset of A002808, A007916, and A106543.
%Y A385410 Cf. A384714, A385411, A385412.
%K A385410 nonn,base,easy
%O A385410 1,1
%A A385410 _Stefano Spezia_ and _Michael De Vlieger_, Jun 28 2025