A013918 Primes equal to the sum of the first k primes for some k.

2, 5, 17, 41, 197, 281, 7699, 8893, 22039, 24133, 25237, 28697, 32353, 37561, 38921, 43201, 44683, 55837, 61027, 66463, 70241, 86453, 102001, 109147, 116533, 119069, 121631, 129419, 132059, 263171, 287137, 325019, 329401, 333821, 338279, 342761

Offset: 1

Author

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

Keywords

• nonn

Comments

Intersection of A000040 and A007504. - David W. Wilson, May 11 2007

Sum of the first k primes p_1+p_2+...+p_k is in the sequence if and only if there exists the prime q for which p_i divides p_1+p_2+...+p_k+q for all i to k. - Vladimir Letsko, Oct 13 2013

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Vladimir Letsko, Mathematical Marathon, problem 124 (in Russian).
• 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. A013916, A013917, A189153 (number of these primes < 10^n).

Cf. A007504, A010051, A000040.

Programs

Haskell

a013918 n = a013918_list !! (n-1)
a013918_list = filter ((== 1) . a010051) a007504_list
-- Reinhard Zumkeller, Feb 09 2015

Mathematica

Select[Accumulate[Prime[Range[1000]]], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

PARI

n=0;forprime(k=2,2300,n=n+k;if(isprime(n),print(n))) \\ Michael B. Porter, Jan 29 2010

Python

from sympy import isprime, nextprime; m = p = 0
while p < 2280:
    p = nextprime(p); m += p
    if isprime(m): print(m, end = ', ') # Ya-Ping Lu, Dec 24 2024

Formula

a(n) = A007504(A013916(n)).

Extensions

More terms from David W. Wilson