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

Author

Gonzalo Martínez, Jun 20 2025

Keywords

• nonn
• base

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.

Examples

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

Crossrefs

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