cp's OEIS Frontend

A385412 Numbers k that are not perfect powers and whose trailing digits form a power of b, where 1 < b < k.

%I A385412 #10 Jul 05 2025 05:15:16
%S A385412 11,12,13,14,15,17,18,19,21,22,23,24,26,28,29,31,33,34,35,37,38,39,41,
%T A385412 42,43,44,45,46,47,48,51,52,53,54,55,56,57,58,59,61,62,63,65,66,67,68,
%U A385412 69,71,72,73,74,75,76,77,78,79,82,83,84,85,86,87,88,89,91,92
%N A385412 Numbers k that are not perfect powers and whose trailing digits form a power of b, where 1 < b < k.
%H A385412 Michael De Vlieger, <a href="/A385412/b385412.txt">Table of n, a(n) for n = 1..10000</a>
%e A385412 Numbers having a single digit are not terms:
%e A385412   1 is not a term since 1 is a power of all bases b;
%e A385412   Numbers k=2..9 are not in the sequence since k^e mod 10 <> b, with e > 1.
%e A385412 10 is not a term since it ends in a single zero, and zero is not a power of another number.
%e A385412 a(1) = 11 since it is a prime and 11 mod 10 is a power of all bases b.
%e A385412 a(2) = 12 since it is not a perfect power, and 12 mod 10 is a power of 2.
%e A385412 a(3) = 13 since it is a prime and 13 mod 10 is a power of 3.
%e A385412 All primes p greater than 7 are terms since they are not perfect powers.
%e A385412 20 is not a term because it ends with a zero, and zero is not a power of another number.
%e A385412 26 is a term since it is not a perfect power, and 26 mod 10 = 6^1.
%t A385412 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 &]
%Y A385412 Union of A385410 and A385411.
%Y A385412 Subset of A007916.
%Y A385412 Cf. A384714.
%K A385412 nonn,base,easy
%O A385412 1,1
%A A385412 _Stefano Spezia_ and _Michael De Vlieger_, Jun 30 2025