A045537 -id:A045537 - cp's OEIS frontend

A175394 Least nontrivial exponent e > 2 such that n^2 is a substring of n^e (n >= 0).

3, 3, 6, 6, 7, 3, 7, 6, 22, 11, 3, 13, 26, 54, 123, 27, 27, 40, 100, 43, 6, 43, 54, 42, 12, 3, 37, 43, 9, 37, 6, 19, 102, 102, 43, 96, 83, 45, 67, 34, 12, 128, 168, 102, 182, 44, 152, 104, 184, 52, 3, 17, 35, 75, 164, 67, 127, 22, 134, 98, 7, 124, 117, 146, 77, 146, 156, 87

Offset: 0

Zak Seidov, Apr 29 2010

			a(2)=6 because 2^2=4 is a substring of 2^6=64
a(4)=7 because 4^2=16 is a substring of 4^7=16384.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A008856, A033819, A045537, A074250.

f:= proc(n) local pat,e;
  pat:= sprintf("%d",n^2);
  for e from 3 do if StringTools:-Search(pat, sprintf("%d",n^e))<> 0 then return e fi od:
end proc:
map(f, [$0..100]); # Robert Israel, Mar 20 2018

lne[n_]:=Module[{e=3,idn2=IntegerDigits[n^2]},While[!MemberQ[ Partition[ IntegerDigits[n^e], Length[ idn2],1],idn2],e++];e]; Array[lne,70,0] (* Harvey P. Dale, Aug 17 2013 *)

A226219 Least k > 1 not a power of 10 such that n is a substring of k*n.

Original entry on oeis.org

2, 11, 6, 11, 6, 3, 6, 11, 6, 11, 11, 101, 26, 87, 51, 21, 26, 69, 51, 63, 6, 58, 51, 97, 26, 5, 51, 101, 26, 79, 11, 101, 26, 101, 51, 21, 26, 101, 51, 87, 6, 59, 34, 101, 26, 21, 51, 74, 26, 51, 3, 69, 26, 29, 51, 21, 26, 101, 51, 27, 6, 92, 51, 26, 26, 21

Offset: 0

Paul Tek, May 31 2013

Paul Tek, Table of n, a(n) for n = 0..10000

Cf. A045537.

import Data.List (isInfixOf, isPrefixOf)
a226219 n = head [k | k <- [2..],
                      isInfixOf (show n) (show (k*n)), not $ p10 k]
   where p10 = flip isPrefixOf ('1' : repeat '0') . show  :: Int -> Bool
-- Reinhard Zumkeller, May 31 2013

Table[k = 2; While[IntegerQ[Log[10, k]] || StringPosition[ToString[k*n], ToString[n], 1] == {}, k++]; k, {n, 0, 100}] (* T. D. Noe, May 31 2013 *)

sub a {
    my $n = shift;
    my $k = 2;
    while (1) {
        if ($k !~ /^10+$/) {
            if (index($k*$n, $n)>=0) {
                return $k;
            }
        }
        $k++;
    }
}

a(n) <= 10^A055642(n)+1.

A343594 Numbers k that, when written in all bases from base 2 to base 10, are a substring of k^k when written in the same base.

Original entry on oeis.org

1, 5, 17, 25, 31, 41, 63, 92, 151, 170, 202, 221, 263, 266, 278, 322, 327, 347, 364, 401, 404, 412, 421, 423, 437, 467, 470, 482, 490, 498, 501, 515, 519, 543, 558, 578, 590, 612, 623, 636, 646, 647, 671, 683, 685, 705, 707, 717, 718, 726, 764, 785, 795, 859, 867, 872, 875, 881, 890, 892, 897

Offset: 1

Scott R. Shannon, Apr 21 2021

			5 is a term. See below table:
.
   base  |  5 in base  |  5^5 in base
---------+-------------+-------------
    10          5                3125
     9          5                4252
     8          5                6065
     7          5               12053
     6          5               22245
     5         10              100000
     4         11              300311
     3         12            11021202
     2        101        110000110101
.
5^5 in all bases contains 5 in that base as a substring.

Cf. A049329, A045537, A051248.

str(v) = my(s=""); for (k=1, #v, s = concat(s, Str(v[k]))); s;
isok(k) = {for (b=2, 10, my(kb = digits(k, b), kkb = digits(k^k, b)); if (#strsplit(str(kkb), str(kb)) <=1 , return (0));); return (1);} \\ Michel Marcus, Apr 26 2021

from sympy.ntheory import digits
def nstr(n, b): return "".join(map(str, digits(n, b=b)[1:]))
def ok(k): return all(nstr(k, b) in nstr(k**k, b) for b in range(10, 1, -1))
print(list(filter(ok, range(900)))) # Michael S. Branicky, Apr 25 2021

cp's OEIS Frontend

A175394 Least nontrivial exponent e > 2 such that n^2 is a substring of n^e (n >= 0).

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Maple

Mathematica

A226219 Least k > 1 not a power of 10 such that n is a substring of k*n.

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Haskell

Mathematica

Perl

Formula

A343594 Numbers k that, when written in all bases from base 2 to base 10, are a substring of k^k when written in the same base.

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PARI

Python