A073683 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), ... This is the sequence of the leading element in each group.

2, 5, 13, 31, 43, 67, 79, 97, 131, 179, 199, 257, 293, 313, 359, 389, 409, 431, 443, 467, 491, 509, 541, 571, 601, 991, 1163, 1523, 1549, 1607, 1627, 1723, 1747, 1787, 1831, 1873, 1907, 2039, 2243, 2269, 2287, 2333, 2347, 2389, 2459, 2521, 2543, 2557, 2593

Offset: 1

Author

Amarnath Murthy, Aug 11 2002

Keywords

• nonn

Comments

First prime of n-th group of successive primes in A073684.

Examples

			Partition the sequence of primes into groups so that the sum of the terms in each group is prime: {2, 3}, {5, 7, 11}, {13, 17, 19, 23, 29}, {31, 37, 41}, {43, 47, 53, 59, 61}, {67, 71, 73}, {79, 83, 89}, {97, 101, 103, 107, 109, 113, 127}, {131, 137, 139, 149, 151, 157, 163, 167, 173}, {179, 181, 191, 193, 197},..; A073684(n) is the number of terms in n-th group; A073682(n) is the sum of terms in n-th group; a(n) is the first term in n-th group; A077279(n) is the last term in n-th group.
		

Crossrefs

Cf. A073682, A073684, A077279.

Programs

Python

from itertools import count, islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    s, i, p = 0, 1, 2
    while True:
        pp = p
        while not(isprime(s:=s+p)) or i < 2:
            i, p = i+1, nextprime(p)
        yield pp
        s, i, p = 0, 1, nextprime(p)
print(list(islice(agen(), 49))) # Michael S. Branicky, May 23 2025

Extensions

More terms from Zak Seidov, Nov 02 2002