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A357190 a(n) is the least prime p such that A234575(p, A007953(p)) is the n-th power of a prime.

%I A357190 #57 Nov 09 2022 19:10:08
%S A357190 17,13,131,107,383,613,43607,1021,334403,26099,40637,138967,212867,
%T A357190 360049,502210997,2227399,5682166613,7339303,13630913,35650627,
%U A357190 92273957,142605709,4424729404133,671087119,42364430471219,2684353351,404156666702231,10737417109,4872756792902003
%N A357190 a(n) is the least prime p such that A234575(p, A007953(p)) is the n-th power of a prime.
%C A357190 a(n) is the least prime p such that the sum of the quotient and remainder on division of p by the sum of digits of p is the n-th power of an integer.
%H A357190 Robert Israel, <a href="/A357190/b357190.txt">Table of n, a(n) for n = 1..100</a>
%e A357190 a(3) = 131 because 131 is prime, has sum of digits 5, 131 = 26*5 + 1 and 26 + 1 = 27 = 3^3 where 3 is prime; and 131 is the least prime that works.
%p A357190 g:= proc(t,M) local s,q,r,n;
%p A357190   for s from 2 to 9*M do
%p A357190     for r from s-1 to 1 by -1 do
%p A357190       q:= t-r;
%p A357190       n:= q*s+r;
%p A357190       if convert(convert(n,base,10),`+`) = s and isprime(n) then return n fi;
%p A357190       if n >= 10^M then return -1 fi;
%p A357190    od od;
%p A357190    -1
%p A357190 end proc:
%p A357190 G:= proc(m) local i,M,found,v,r;
%p A357190   found:= false; r:= infinity;
%p A357190   for M from 3 while not found do
%p A357190     for i from 1 while ithprime(i)^m < 10^M do
%p A357190       v:= g(ithprime(i)^m, M);
%p A357190       if v > 0 then found:= true; r:= min(v,r) fi
%p A357190   od od:
%p A357190   r
%p A357190 end proc:
%p A357190 map(G, [$1..30]);
%Y A357190 Cf. A007953, A234575.
%K A357190 nonn,base
%O A357190 1,1
%A A357190 _J. M. Bergot_ and _Robert Israel_, Oct 25 2022