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

1,2

Links

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

Example

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.

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 *)

Crossrefs

Cf. A001414, A001597, A356646.

Sequence in context: A132708 A153727 A349245 * A210937 A016506 A346995

Adjacent sequences: A356628 A356629 A356630 * A356632 A356633 A356634

Keyword

nonn

Author

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

Status

approved