A074250 -id:A074250 - 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 *)

A378857 a(n) is the smallest k > 1 such that n^k starts and ends with n, or -1 if there is no such k.

Original entry on oeis.org

2, 21, 41, 11, 24, 10, 33, 73, 153, -1, 171, 241, 361, -1, -1, 6, 461, -1, 471, -1, 12086, -1, 1281, 51, 94, -1, 1181, 701, 1091, -1, 231, 197, 5781, -1, -1, 161, 221, -1, 1231, -1, 236, -1, 61, 1451, -1, -1, 861, 13381, 143, -1, 107, 501, 2761, -1, -1, 136

Offset: 1

Gonzalo Martínez, Feb 10 2025

Given an integer n such that a(n) != -1, it follows that n^A051248(n) starts with n and n^A074250(n) ends with n, and both are the smallest exponents that satisfy these conditions. In this case, n^a(n) starts and ends with n. Due to this condition, larger exponents are required. For example, for n = 21 the smallest exponent satisfying the condition is 12086.

			2^21 = 2097152 is the smallest power of 2 that starts and ends with 2, so a(2) = 21.

Cf. A051248, A074250, A374026.

a(n) >= max{A074250(n), A051248(n)} except that a(n) = -1 when A074250(n) = -1.

A385171 Perfect powers m^k whose decimal expansion begins with k and ends with m, where m and k are greater than 1.

Original entry on oeis.org

25, 59049, 78125, 13060694016, 17179869184, 19073486328125, 30514648531249, 53613724194557, 59120987373568, 65944160601201, 116490258898219, 324965351768751, 512908935546875, 21936950640377856, 371308922853718751, 578261433548013568, 913517247483640899

Offset: 1

Gonzalo Martínez, Jun 20 2025

Such as automorphic numbers (A003226), which are those m such that m^2 ends with m, if m^k is in this sequence, then it is a k-morphic number which also begins with k. Thus, m^k contains both m and k as substrings at its ends.
If m is in A003226 and m^2 starts with 2, then m^2 is in this sequence. For example, A003226(3)^2 = 5^2 = 25 and A003226(119)^2.
If m is in A033819 and m^3 starts with 3, then m^3 is in this sequence. For example, A033819(39)^3 = 31249^3 = 30514648531249.
This sequence has infinitely many terms since (10^m - 1)^9 is a term for all m >= 2, which starts with (m - 1) 9's and ends with m 9's.

			6^13 = 13060694016 is a term since it starts with 13 and ends with 6.

Cf. A001597, A003226, A033819, A378857, A051248, A074250.

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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A378857 a(n) is the smallest k > 1 such that n^k starts and ends with n, or -1 if there is no such k.

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A385171 Perfect powers m^k whose decimal expansion begins with k and ends with m, where m and k are greater than 1.

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