cp's OEIS Frontend

A356631 a(n) is the least number k such that the sum (with multiplicity) of prime factors of k*(k+1)*...*(k+n-1) is a perfect power.

%I A356631 #12 Sep 05 2022 09:10:32
%S A356631 1,4,2,1,4,5,2,1,11,18,8,12,8,15,4,41,10,65,10,39,21,5,54,30,25,2,1,
%T A356631 17,43,2,1,80,12,41,206,11,70,39,81,5,289,50,18,56,24,10,49,103,146,
%U A356631 77,53,582,31,58,37,419,140,174,77,44,100,168,44,42,99,13,11,80,60,101,71,12,24,70,11,52,671
%N A356631 a(n) is the least number k such that the sum (with multiplicity) of prime factors of k*(k+1)*...*(k+n-1) is a perfect power.
%H A356631 Robert Israel, <a href="/A356631/b356631.txt">Table of n, a(n) for n = 1..10000</a>
%e A356631 a(5) = 4 because the sum of prime factors of 4*5*6*7*8 = 2^6 * 3 * 5 * 7 is 2*6 + 3 + 5 + 7 = 27 = 3^3 is a perfect power, and 4 is the least number that works.
%p A356631 spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
%p A356631 ispow:= proc(n) igcd(map(t -> t[2], ifactors(n)[2]))>1 end proc:
%p A356631 f:= proc(n) local S,t,i;
%p A356631   S:= Vector(n,spf); t:= convert(S,`+`);
%p A356631     for i from 1 do
%p A356631       if ispow(t) then return i fi;
%p A356631     t:= t-S[1];
%p A356631     S[1..n-1]:= S[2..n];
%p A356631     S[n]:= spf(i+n);
%p A356631     t:= t+S[n];
%p A356631   od
%p A356631 end proc:
%p A356631 f(1):= 1:
%p A356631 map(f, [$1..100]);
%t A356631 sopfr[n_] := Plus @@ Times @@@ FactorInteger[n]; powQ[n_] := GCD @@ FactorInteger[n][[;; , 2]] > 1; a[n_] := Module[{k = 1}, While[! powQ[sopfr[Product[k + i, {i, 0, n - 1}]]], k++]; k]; a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Aug 19 2022 *)
%Y A356631 Cf. A001414, A001597, A356646.
%K A356631 nonn
%O A356631 1,2
%A A356631 _J. M. Bergot_ and _Robert Israel_, Aug 18 2022