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

%I A052349 #48 Dec 12 2022 15:00:05
%S A052349 1,8,24,25,86,1260,1890,14136,197400,10467660,1231572090,682616834970
%N A052349 Lexicographically earliest sequence of distinct positive integers such that no subsequence sums to a prime.
%C A052349 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.
%H A052349 Chris Nash, <a href="/A052349/a052349.txt">Proof that A052349, A128687, and A128688 are infinite</a> [Cached copy of proof, from The Prime Puzzles and Problems website]
%H A052349 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_084.htm">Puzzle 84. Non-primes adding up to non-primes</a>, The Prime Puzzles and Problems Connection.
%e A052349 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.
%t A052349 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 *)
%o A052349 (Python)
%o A052349 from sympy import isprime
%o A052349 from itertools import islice
%o A052349 def agen(start=1): # generator of terms
%o A052349     alst, k, sums = [start], 1, {0} | {start}
%o A052349     while True:
%o A052349         yield alst[-1]
%o A052349         while any(isprime(k + s) for s in sums): k += 1
%o A052349         alst.append(k)
%o A052349         sums.update([k + s for s in sums])
%o A052349         k += 1
%o A052349 print(list(islice(agen(), 9))) # _Michael S. Branicky_, Dec 12 2022
%Y A052349 Cf. A068638.
%Y A052349 Cf. A128687 (restricted to odd numbers), A128688 (restricted to even numbers).
%K A052349 hard,more,nonn,nice
%O A052349 1,2
%A A052349 _Carlos Rivera_, Mar 07 2000
%E A052349 One more term from _T. D. Noe_, Mar 20 2007
%E A052349 a(12) from _Donovan Johnson_, Jun 26 2010
%E A052349 New name from _Charles R Greathouse IV_, Jan 13 2014
%E A052349 Name clarified by _Peter Kagey_, Jan 07 2017