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.

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, 17, 43, 2, 1, 80, 12, 41, 206, 11, 70, 39, 81, 5, 289, 50, 18, 56, 24, 10, 49, 103, 146, 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

Offset: 1

Author

J. M. Bergot and Robert Israel, Aug 18 2022

Keywords

• nonn

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001414, A001597, A356646.

Programs

Maple

spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
ispow:= proc(n) igcd(map(t -> t[2], ifactors(n)[2]))>1 end proc:
f:= proc(n) local S,t,i;
  S:= Vector(n,spf); t:= convert(S,`+`);
    for i from 1 do
      if ispow(t) then return i fi;
    t:= t-S[1];
    S[1..n-1]:= S[2..n];
    S[n]:= spf(i+n);
    t:= t+S[n];
  od
end proc:
f(1):= 1:
map(f, [$1..100]);

Mathematica

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 *)