

A049329


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


9



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
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OFFSET

1,2


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



MAPLE

filter:= proc(n) local L, Ln;
L:= convert(n, string);
Ln:= convert(n^n, string);
StringTools:Search(L, Ln) <> 0
end proc:


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 *)


CROSSREFS



KEYWORD

base,nonn,easy,nice,changed


AUTHOR



STATUS

approved



