A061280 -id:A061280 - cp's OEIS frontend

A239134 Smallest k such that n^k contains k as a substring in its decimal representation.

1, 6, 7, 6, 2, 6, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 4, 2, 3, 2, 4, 2, 4, 3, 7, 1, 2, 3, 3, 2, 2, 3, 5, 2, 6, 1, 8, 4, 4, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 4, 2, 4, 3, 6, 1, 3, 5, 6, 2, 4, 3, 2, 3, 3, 1, 3, 2, 6, 2, 3, 2, 6, 2, 4, 1, 2, 4, 4

Offset: 1

Derek Orr, Mar 10 2014

It seems very likely a(n) < 10 for all n (even stronger, a(n) < 9 for all n).
It also seems very likely a(n) = {1,2,3} for sufficiently large n.
Counterexample: a(10^d - 2) = 6 for d >= 2. - Robert Israel, Sep 16 2024

			5^1 = 5 does not contain a 1 but 5^2 = 25 does contain a 2 so a(5) = 2.
7^1 = 7 does not contain a 1, 7^2 = 49 does not contain a 2, but 7^3 = 343 does contain a 3 so a(7) = 3.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A045537, A061280.

f:= proc(n) local k;
  for k from 1 to 9 do
    if member(k,convert(n^k,base,10)) then return k fi
  od;
  FAIL
end proc:
map(f, [$1..100]); # Robert Israel, Sep 16 2024

a[n_] := Block[{k=1}, While[{} == StringPosition[ ToString[n^k], ToString[k]], k++]; k]; Array[a, 84] (* Giovanni Resta, Mar 11 2014 *)
sk[n_]:=Module[{k=1},While[SequenceCount[IntegerDigits[n^k],IntegerDigits[k]] == 0,k++];k]; Array[sk,90] (* Harvey P. Dale, May 12 2022 *)

def Sub(x):
  for n in range(10**3):
    if str(x**n).find(str(n)) > -1:
      return n
x = 1
while x < 10**3:
  print(Sub(x))
  x += 1

a(A011531(k))=1, any k.

a(10*n) = a(n) if a(n) < 10. - Robert Israel, Sep 16 2024

A277079 Least number k for which k^n contains n as a substring and n^k contains k as a substring.

Original entry on oeis.org

1, 35, 7, 61, 5, 6, 3, 7, 5, 10, 11, 15, 6, 36, 7, 16, 17, 33, 16, 44, 6, 48, 36, 3, 5, 9, 31, 8, 32, 69, 8, 5, 8, 5, 2, 2, 9, 8, 6, 6, 5, 8, 7, 6, 6, 9, 8, 6, 2, 7, 2, 8, 6, 5, 5, 5, 3, 8, 9, 6, 4, 3, 6, 6, 6, 25, 5, 6, 3, 6, 3, 3, 2, 6, 5, 6, 3, 7, 8, 7, 4, 2

Offset: 1

Paolo P. Lava, Sep 28 2016

a(n) = n for 1, 5, 6, 10, 11, 16, 17.

Records for a(1) = 1, a(2) = 35, a(4) = 61, a(30) = 69, a(1061) = 84, a(1256) = 91, ...

			2^35 = 34359738368 and 35 is a substring;
35^2 = 1225 and 2 is a substring.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A061280.

P:=proc(q) local a,b,j,k,n,ok; for n from 1 to q do a:=convert(n,string); ok:=1; for k from 1 to q do if ok=1 then if searchtext(a,convert(k^n,string))>0 then b:=convert(k,string);
for j from 1 to q do if searchtext(b,convert(n^k,string))>0 then print(k); ok:=0; break; fi; od; fi; fi; od; od; end: P(10^3);

Table[k = 1; While[Or[Length@ SequencePosition[IntegerDigits[k^n], IntegerDigits[n]] == 0, Length@ SequencePosition[IntegerDigits[n^k], IntegerDigits[k]] == 0], k++]; k, {n, 120}] (* Michael De Vlieger, Sep 28 2016, Version 10.1 *)

cp's OEIS Frontend

A239134 Smallest k such that n^k contains k as a substring in its decimal representation.

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