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A052349
Lexicographically earliest sequence of distinct positive integers such that no subsequence sums to a prime.
15
1, 8, 24, 25, 86, 1260, 1890, 14136, 197400, 10467660, 1231572090, 682616834970
OFFSET
1,2
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.
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.
EXAMPLE
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.
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]&&d<Length@s, d++]; d!=Length@s||PrimeQ[Total[s[[d]]]+c]||PrimeQ@c, c++]; c); Array[a, 8] (* Giorgos Kalogeropoulos, Nov 19 2021 *)
PROG
(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
CROSSREFS
Cf. A068638.
Cf. A128687 (restricted to odd numbers), A128688 (restricted to even numbers).
Sequence in context: A370531 A109272 A361998 * A254448 A029607 A244030
KEYWORD
hard,more,nonn,nice
AUTHOR
Carlos Rivera, Mar 07 2000
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
STATUS
approved