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

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

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..17.

Example

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

Crossrefs

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

Sequence in context: A203543 A337727 A034711 * A325215 A076445 A013835

Adjacent sequences: A385168 A385169 A385170 * A385172 A385173 A385174

Keyword

nonn,base

Author

Gonzalo Martínez, Jun 20 2025

Status

approved