A385410 Multiples k of b that are not perfect powers and whose trailing digits form a power of b, where 1 < b < k.
12, 14, 15, 18, 21, 22, 24, 28, 33, 34, 35, 38, 39, 42, 44, 45, 48, 51, 52, 54, 55, 58, 62, 63, 65, 66, 68, 69, 72, 74, 75, 77, 78, 82, 84, 85, 88, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105, 108, 110, 111, 112, 114, 115, 116, 118, 120, 122, 123, 124, 126, 129
Offset: 1
Examples
In general, no prime p is a term since they are a power of base p. Numbers having a single digit are not terms: 1 is not a term since 1 is a power of all bases b; Composites 4, 6, and 9 are not in the sequence since 4 = 2^2, 6 = 2*3, and 9 = 3^2. 10 is not a term since it ends in a single zero, and zero is not a power of another number. a(1) = 12 since it is not a perfect power, 2 | 12, and 12 mod 10 is a power of 2. a(2) = 14 since it is not a perfect power, 2 | 14, and 14 mod 10 is a power of 2. 20 is not a term since it ends with a zero, and zero is not a power of another number. 26 is not a term since 6 does not divide 26. 1100 is a term since it is not a perfect power, 100 = 100^1, and 100 | 1100.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
Programs
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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[And[Divisible[k, b], FreeQ[s, k]], Sow[k]]; i++]) &, s] ][[-1]]], {b, 2, nn}]; Select[t, GCD @@ FactorInteger[#][[;; , -1]] === 1 &]