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

%I A346067 #18 Aug 08 2021 16:28:01
%S A346067 2,37,17,7,109,36013476739,31,80051,71,97,107,13093,103,127,107,163,
%T A346067 991,181,157,181,199,193,271,31663,211,307,307,5318989651,673,8297,
%U A346067 331,811,359,463
%N A346067 Smallest prime that is the n-th power analog of Keith numbers.
%C A346067 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.
%C A346067 The only terms for n <= 100 whose values are still unknown are a(35), a(90), a(91) and a(95).
%C A346067 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.
%H A346067 Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_1043.htm">Puzzle 1043. Another puzzle about Keith numbers</a>, The Prime Puzzles & Problems Connection.
%e A346067 a(2) = 37 because 37^2 = 1369. Then 1+3+6+9 = 19 and 3+6+9+19 = 37.
%t A346067 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];
%t A346067 KeithPow[n_]:=(k=1;While[!KeithPowQ[Prime@k,n],k++];Prime@k);Array[KeithPow,5] (* code modified from A007629 *)
%Y A346067 Cf. A007629 (Keith numbers).
%Y A346067 Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281919, A281920, A281921 (starting with n^k, 2<=k<=10).
%K A346067 nonn,base,more
%O A346067 1,1
%A A346067 _Giorgos Kalogeropoulos_, Jul 03 2021