A013916 Numbers k such that the sum of the first k primes is prime.

1, 2, 4, 6, 12, 14, 60, 64, 96, 100, 102, 108, 114, 122, 124, 130, 132, 146, 152, 158, 162, 178, 192, 198, 204, 206, 208, 214, 216, 296, 308, 326, 328, 330, 332, 334, 342, 350, 356, 358, 426, 446, 458, 460, 464, 480, 484, 488, 512, 530, 536, 548, 568, 620, 630, 676, 680

Offset: 1

Author

N. J. A. Sloane, Renaud Lifchitz (100637.64(AT)CompuServe.COM)

Keywords

• nonn
• nice

Examples

			6 is a term because the sum of the first six primes 2 + 3 + 5 + 7 + 11 + 13 = 41 is prime.
		

Links

• David W. Wilson, Table of n, a(n) for n = 1..10000
• Romeo Meštrović, Curious conjectures on the distribution of primes among the sums of the first 2n primes, arXiv:1804.04198 [math.NT], 2018.

Crossrefs

Cf. A007504, A013917, A013918.

Programs

GAP

P:=Filtered([1..5300],IsPrime);;
a:=Filtered([1..Length(P)],n->IsPrime(Sum([1..n],k->P[k])));; Print(a); # Muniru A Asiru, Jan 04 2019

MATLAB

p=primes(10000); m=1;
for u=1:700 ; suma=sum(p(1:u));
     if isprime(suma)==1 ; sol(m)=u; m=m+1; end
end
sol; % Marius A. Burtea, Jan 04 2019

Magma

[n:n in [1..700] | IsPrime(&+PrimesUpTo(NthPrime(n))) ]; // Marius A. Burtea, Jan 04 2019

Maple

p:=proc(n) if isprime(sum(ithprime(k),k=1..n))=true then n else fi end: seq(p(n),n=1..690); # Emeric Deutsch

Mathematica

s = 0; Do[s = s + Prime[n]; If[PrimeQ[s], Print[n]], {n, 1, 1000}]
Flatten[Position[Accumulate[Prime[Range[2000]]], ?(PrimeQ[#] &)]] (* _Harvey P. Dale, Dec 16 2010 *)
Flatten[Position[PrimeQ[Accumulate[Prime[Range[2000]]]],True]] (* Fred Patrick Doty, Aug 15 2017 *)

PARI

isA013916(n) = isprime(sum(i=1,n,prime(i))) \\ Michael B. Porter, Jan 29 2010

Python

from sympy import isprime, prime
def aupto(lim):
  s = 0
  for k in range(1, lim+1):
    s += prime(k)
    if isprime(s): print(k, end=", ")
aupto(680) # Michael S. Branicky, Feb 28 2021

Formula

a(n) = A000720(A013917(n)).

Extensions

More terms from David W. Wilson