A302567 a(n) is the number of primes less than the n-th prime that divide the sum of primes up to the n-th prime.

0, 0, 1, 0, 2, 0, 1, 2, 2, 1, 2, 0, 3, 0, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 2, 2, 1, 3, 2, 3, 1, 3, 1, 1, 1, 3, 2, 3, 2, 3, 1, 3, 1, 3, 1, 2, 2, 3, 3, 3, 2, 4, 1, 1, 3, 4, 2, 1, 0, 2, 1, 2, 0, 1, 2, 2, 3, 2, 3, 3, 1, 3, 1, 1, 2, 4, 1, 3, 3, 1, 1, 1, 4, 3, 2, 4, 3, 3, 3, 4, 1, 1, 2, 1, 0, 2, 3, 2, 0, 2, 0, 4, 1, 4

Offset: 1

Author

G. L. Honaker, Jr., Apr 11 2018

Keywords

• nonn

Comments

This sequence differs from A105783 only at n = 1, 3, 20, 31464, 22096548, ... (the terms of A024011); see Example section. - Jon E. Schoenfield, Apr 11 2018

Examples

			a(13)=3 because the 13th prime is 41 and the sum of primes up to 41 is 238, which has 3 distinct prime factors less than 41.
a(20)=1 because the 20th prime is 71 and the sum of primes up to 71 is 639 = 7*71, which has only 1 distinct prime factor less than 71. - _Jon E. Schoenfield_, Apr 11 2018
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Caldwell and Honaker, Prime Curios!: 163117

Crossrefs

Cf. A000040, A007504, A024011, A105783.

Programs

Maple

s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
a:= n-> nops(select(x-> x < ithprime(n), numtheory[factorset](s(n)))):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 11 2018

Mathematica

a[n_] := (S = Total[P = Prime[Range[n]]]; Count[P, p_ /; Divisible[S, p]]);
Array[a, 100] (* Jean-François Alcover, Apr 30 2019 *)

PARI

a(n) = #select(x->(x < prime(n)), factor(sum(k=1, n, prime(k)))[,1]); \\ Michel Marcus, Apr 11 2018

Formula

a(n) = A105783(n) - 1 if n is in A024011; otherwise, a(n) = A105783(n). - Jon E. Schoenfield, Apr 11 2018