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

1,1

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.

Links

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

Example

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.

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 *)

Prog

(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

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.

Sequence in context: A001269 A201769 A077276 * A179295 A083776 A122820

Adjacent sequences: A073681 A073682 A073683 * A073685 A073686 A073687

Keyword

nonn

Author

Amarnath Murthy, Aug 11 2002

Extensions

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

Extended by Ray Chandler, May 02 2007

Status

approved