A346067 Smallest prime that is the n-th power analog of Keith numbers.

2, 37, 17, 7, 109, 36013476739, 31, 80051, 71, 97, 107, 13093, 103, 127, 107, 163, 991, 181, 157, 181, 199, 193, 271, 31663, 211, 307, 307, 5318989651, 673, 8297, 331, 811, 359, 463

Offset: 1

Author

Giorgos Kalogeropoulos, Jul 03 2021

Keywords

• nonn
• base
• more

Comments

The n-th power analog of Keith numbers is like Keith numbers but starting from p^n to reach p. Consider the digits of p^n where p is prime. Take their sum and repeat the process, deleting the first addend and adding the previous sum. We are searching for the first prime p that after some number of iterations reaches a sum equal to p.

The only terms for n <= 100 whose values are still unknown are a(35), a(90), a(91) and a(95).

Paolo Lava asked for these numbers as a puzzle (see the Rivera link) and as a result a(61) = 11659149703 and a(81) = 200908021 were found.

Examples

			a(2) = 37 because 37^2 = 1369. Then 1+3+6+9 = 19 and 3+6+9+19 = 37.
		

Links

• Carlos Rivera, Puzzle 1043. Another puzzle about Keith numbers, The Prime Puzzles & Problems Connection.

Crossrefs

Cf. A007629 (Keith numbers).

Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281919, A281920, A281921 (starting with n^k, 2<=k<=10).

Programs

Mathematica

KeithPowQ[m_Integer,n_]:=Module[{b=IntegerDigits[m^n],s,k=0},s=Total[b];While[sA007629 *)