A105783 -id:A105783 - cp's OEIS frontend

A102863 a(n)=1 if at least one of the first n primes is a divisor of the sum of the first n primes; otherwise a(n)=0.

1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1

Offset: 1

Giovanni Teofilatto, Mar 01 2005

a(n) = 0 if and only if n is in A013916. - Robert Israel, Jan 04 2017

			a(2)=0 because none of the first 2 primes (2, 3) is a divisor of 2+3; a(5)=1 because among the first 5 primes (namely, 2,3,5,7,11) there are divisors of 2+3+5+7+11=28.

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

A105783(n) gives number of primes among the first n primes that are divisors of the sum of the first n primes.

Cf. A013916, A136443.

with(numtheory):
a:=proc(n)
   if nops(factorset(sum(ithprime(k),k=1..n)) intersect {seq(ithprime(j),j=1..n)}) >0 then
      1
   else
      0
   fi
end:
seq(a(n),n=1..130); # Emeric Deutsch
# alternative:
N:= 500: # to get the first N terms
A:= Vector(N):
S:= 2: P:= 2: p:= 2: A[1]:= 1:
for n from 2 to N do
  p:= nextprime(p);
  S:= S+p; P:= P*p;
  if igcd(S,P) > 1 then A[n]:= 1 fi
od:
convert(A,list); # Robert Israel, Jan 04 2017

a[n_] := Module[{pp = Prime[Range[n]], t}, t = Total[pp]; Boole[AnyTrue[pp, Divisible[t, #]&]]];
Array[a, 100] (* Jean-François Alcover, Jun 16 2020 *)

Edited and extended by Emeric Deutsch, Apr 19 2005

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.

Original entry on oeis.org

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

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

nonn

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

			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

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

Cf. A000040, A007504, A024011, A105783.

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

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

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

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

cp's OEIS Frontend

A102863 a(n)=1 if at least one of the first n primes is a divisor of the sum of the first n primes; otherwise a(n)=0.

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