A073684 Sum of next a(n) successive primes is prime.

2, 3, 5, 3, 5, 3, 3, 7, 9, 5, 9, 7, 3, 7, 5, 3, 3, 3, 5, 3, 3, 3, 5, 5, 57, 25, 49, 3, 9, 5, 11, 3, 5, 5, 5, 5, 17, 25, 3, 3, 5, 3, 7, 9, 5, 3, 3, 3, 15, 3, 3, 3, 3, 3, 3, 3, 15, 3, 5, 33, 5, 3, 3, 9, 7, 3, 33, 3, 3, 5, 3, 15, 3, 5, 9, 7, 13, 5, 11, 3, 3, 11

Offset: 1

Author

Amarnath Murthy, Aug 11 2002

Keywords

• nonn

Comments

Group the primes such that the sum of each group is a prime. Each group from the second onwards should contain at least 3 primes: (2, 3), (5, 7, 11), (13, 17, 19, 23, 29), (31, 37, 41), (43, 47, 53, 59, 61), ... Sequence gives number of terms in each group.

Examples

			a(1)=2 because sum of first two primes 2+3 is prime; a(2)=3 because sum of next three primes 5+7+11 is prime; a(3)=5 because sum of next five primes 13+17+19+23+29 is prime.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A073682(n) is the sum of terms in n-th group, A073683(n) is the first term in n-th group, A077279(n) is the last term in n-th group.

Programs

Mathematica

f[l_List] := Block[{n = Length[Flatten[l]], k = 3, r},While[r = Table[Prime[i], {i, n + 1, n + k}]; ! PrimeQ[Plus @@r], k += 2];Append[l, r]];Length /@ Nest[f, {{2, 3}}, 100] (* Ray Chandler, May 11 2007 *)
cnt = 0; Table[s = Prime[cnt+1] + Prime[cnt+2]; len = 2; While[! PrimeQ[s], len++; s = s + Prime[cnt+len]]; cnt = cnt + len; len, {n, 100}] (* T. D. Noe, Feb 06 2012 *)

Python

from itertools import count, islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    s, i, p = 0, 1, 2
    while True:
        while not(isprime(s:=s+p)) or i < 2:
            i, p = i+1, nextprime(p)
        yield i
        s, i, p = 0, 1, nextprime(p)
print(list(islice(agen(), 82))) # Michael S. Branicky, May 23 2025

Extensions

More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Apr 10 2003

Extended by Ray Chandler, May 02 2007