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%I #29 Jan 02 2023 12:30:54
%S 0,1,2,3,4,9,8,10,33,14,93,20,17,23,44,6,24,35,65,5,18,32,11,12,29,30,
%T 7,31,72,16,22,25,37,15,46,64,43,28,85,19,54,13,88,34,49,39,40,27,100,
%U 57,26,52,111,21,38,45,62,41,51,56,47,116,50,81,63,68,59,170,69,71
%N For all n >= 0, six among (a(n+i) + a(n+j), 0 <= i < j < 5) are prime: lexicographically first such sequence of distinct nonnegative integers.
%C The restriction to [1, oo) is the lexicographically first such sequence of positive integers. (This is rather exceptional, cf. A128280 vs A055265, A329405 vs A329450, ..., see the wiki page for more.)
%C Conjectured to be a permutation, i.e., all n >= 0 appear. The restriction to [1, oo) is then the lexicographically first such permutation of the positive integers.
%C Among pairwise sums of 5 consecutive terms, there cannot be more than 2 x 3 = 6 primes: see the wiki page for this and further considerations and variants.
%H Robert Israel, <a href="/A329425/b329425.txt">Table of n, a(n) for n = 0..10000</a>
%H Éric Angelini, <a href="http://list.seqfan.eu/oldermail/seqfan/2019-November/020145.html">Prime sums from neighbouring terms</a>, SeqFan list, and <a href="http://cinquantesignes.blogspot.com/2019/11/prime-sums-from-neighbouring-terms.html">personal blog "Cinquante signes"</a>, Nov. 11, 2019.
%H M. F. Hasler, <a href="/wiki/User:M._F._Hasler/Prime_sums_from_neighboring_terms">Prime sums from neighboring terms</a>, OEIS wiki, Nov. 23, 2019.
%p R:= 0,1,2,3,4:
%p S:= {R}:
%p for i from 1 to 100 do
%p for x from 5 do
%p if member(x,S) then next fi;
%p n1:= nops(select(isprime,[seq(seq(R[i+j]+R[i+k],j=1..k-1),k=1..4)]));
%p if nops(select(isprime,[seq(R[i+j]+x,j=1..4)]))+n1 = 6 then
%p R:= R, x; S:= S union {x}; break
%p fi
%p od od:
%p R; # _Robert Israel_, Dec 29 2022
%o (PARI) A329425_upto(N) = S(N,6,5,0) \\ see the wiki page for the function S().
%Y Cf. A055265, A128280 (1 prime from 2 terms), A329333 (1 prime from 3 terms), A329405-A329416 (N primes from M terms >= 1), A329449, ..., A329581 (N primes from M terms >= 0).
%K nonn
%O 0,3
%A _M. F. Hasler_, following an idea from _Eric Angelini_, Nov 24 2019
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