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

1,1

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.

Links

Table of n, a(n) for n=1..34.

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

Example

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

Mathematica

KeithPowQ[m_Integer, n_]:=Module[{b=IntegerDigits[m^n], s, k=0}, s=Total[b]; While[s<m, AppendTo[b, s]; k++; s=2*s-b[[k]]]; s==m];
KeithPow[n_]:=(k=1; While[!KeithPowQ[Prime@k, n], k++]; Prime@k); Array[KeithPow, 5] (* code modified from A007629 *)

Crossrefs

Cf. A007629 (Keith numbers).

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

Sequence in context: A258035 A178200 A049106 * A290803 A042569 A066196

Adjacent sequences: A346064 A346065 A346066 * A346068 A346069 A346070

Keyword

nonn,base,more

Author

Giorgos Kalogeropoulos, Jul 03 2021

Status

approved