cp's OEIS Frontend

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

%I A385171 #8 Jun 25 2025 11:03:06
%S A385171 25,59049,78125,13060694016,17179869184,19073486328125,30514648531249,
%T A385171 53613724194557,59120987373568,65944160601201,116490258898219,
%U A385171 324965351768751,512908935546875,21936950640377856,371308922853718751,578261433548013568,913517247483640899
%N A385171 Perfect powers m^k whose decimal expansion begins with k and ends with m, where m and k are greater than 1.
%C A385171 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.
%C A385171 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.
%C A385171 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.
%C A385171 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.
%e A385171 6^13 = 13060694016 is a term since it starts with 13 and ends with 6.
%Y A385171 Cf. A001597, A003226, A033819, A378857, A051248, A074250.
%K A385171 nonn,base
%O A385171 1,1
%A A385171 _Gonzalo Martínez_, Jun 20 2025