A049329 Numbers k such that k is a substring of k^k.

1, 5, 6, 9, 10, 11, 16, 17, 19, 21, 24, 25, 28, 31, 32, 33, 35, 36, 37, 39, 41, 42, 43, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 59, 60, 61, 63, 64, 66, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 103, 108, 111, 115

Offset: 1

Author

David W. Wilson

Keywords

• base
• nonn
• easy
• nice

Comments

For typical large k, the string corresponding to k^k has length on the order of k log_10(k); heuristically, each substring of length d = log_10(k) has probability 10^(-d) ~ 1/k of matching k, and the probability that none of these matches is about exp(-log_10(k)) = k^(-log_10(e)) ~ k^(-0.434). Thus we should expect that most large k are in the sequence, but infinitely many are not. - Robert Israel, Jul 14 2015

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000312.

Programs

Maple

filter:= proc(n) local L, Ln;
L:= convert(n,string);
Ln:= convert(n^n,string);
StringTools:-Search(L,Ln) <> 0
end proc:
select(filter, [$1..1000]); # Robert Israel, Jul 13 2015

Mathematica

ssQ[n_] := Module[{idn = IntegerDigits[n]}, MemberQ[Partition[ IntegerDigits[ n^n], Length[idn], 1], idn]]; Select[Range[120], ssQ] (* Harvey P. Dale, Apr 01 2011 *)
Select[Range[120],SequenceCount[IntegerDigits[#^#],IntegerDigits[#]]>0&] (* Harvey P. Dale, Sep 18 2023 *)