A052349 Lexicographically earliest sequence of distinct positive integers such that no subsequence sums to a prime.

1, 8, 24, 25, 86, 1260, 1890, 14136, 197400, 10467660, 1231572090, 682616834970

Offset: 1

Author

Carlos Rivera, Mar 07 2000

Keywords

• hard
• more
• nonn
• nice

Comments

This set was defined by T. W. A. Baumann in The Prime Puzzles and Problems pages. He and C. Rivera obtained the first 10 members. Chris Nash proved that this sequence is infinite.

Examples

			a(4) = 25 as 25+1, 25+8, 25+24, 25+1+8, 25+1+24, 25+8+24 and finally 25+1+8+24 all are composite numbers.
		

Links

• Chris Nash, Proof that A052349, A128687, and A128688 are infinite [Cached copy of proof, from The Prime Puzzles and Problems website]
• Carlos Rivera, Puzzle 84. Non-primes adding up to non-primes, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A068638.

Cf. A128687 (restricted to odd numbers), A128688 (restricted to even numbers).

Programs

Mathematica

a[1]=1; a[n_]:=a[n]=(s=Subsets[Array[a,n-1],n-1]; c=a[n-1]; While[d=1;While[!PrimeQ[Total[s[[d]]]+c]&&dGiorgos Kalogeropoulos, Nov 19 2021 *)

Python

from sympy import isprime
from itertools import islice
def agen(start=1): # generator of terms
    alst, k, sums = [start], 1, {0} | {start}
    while True:
        yield alst[-1]
        while any(isprime(k + s) for s in sums): k += 1
        alst.append(k)
        sums.update([k + s for s in sums])
        k += 1
print(list(islice(agen(), 9))) # Michael S. Branicky, Dec 12 2022

Extensions

One more term from T. D. Noe, Mar 20 2007

a(12) from Donovan Johnson, Jun 26 2010

New name from Charles R Greathouse IV, Jan 13 2014

Name clarified by Peter Kagey, Jan 07 2017