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A329576 For all n >= 1, exactly seven sums are prime among a(n+i) + a(n+j), 0 <= i < j < 6; lexicographically earliest such sequence of distinct positive numbers.

%I A329576 #5 Feb 11 2020 12:50:01
%S A329576 1,2,3,4,5,8,11,26,15,9,14,32,17,20,21,27,10,16,19,7,12,13,24,6,23,35,
%T A329576 25,37,18,36,22,31,61,28,30,39,40,43,33,64,38,45,34,29,63,50,44,53,42,
%U A329576 59,47,54,48,41,90,49,55,52,108,58,46,51,121,73,78,76,100,79,81,151,60,67,112,70,69
%N A329576 For all n >= 1, exactly seven sums are prime among a(n+i) + a(n+j), 0 <= i < j < 6; lexicographically earliest such sequence of distinct positive numbers.
%C A329576 That is, there are 7 primes, counted with multiplicity, among the 15 pairwise sums of any 6 consecutive terms.
%C A329576 Conjectured to be a permutation of the positive integers.
%H A329576 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
%e A329576 For n = 1, we must forbid the greedy choice for a(6) which would be 6, which leads to a dead end: there is no possibility to find a subsequent term that would give 7 prime sums together with {2, 3, 4, 5, 6}. If we take the next larger possibility, a(6) = 8, then it works for the next and all subsequent terms.
%o A329576 (PARI) {A329576(n,show=1,o=1,N=7,M=5,X=[[6,6]],p=[],u=o,U)=for(n=o+1,n, show>0&& print1(o","); show<0&& listput(L,o); U+=1<<(o-u); U>>=-u+u+=valuation(U+1,2); p=concat(if(#p>=M,p[^1],p),o); my(c=N-sum(i=2,#p, sum(j=1,i-1, isprime(p[i]+p[j])))); if(#p<M&&sum(i=1,#p,isprime(p[i]+u))<=c,o=u)|| for(k=u,oo,bittest(U,k-u)|| sum(i=1,#p,isprime(p[i]+k))!=c|| setsearch(X,[n,k])|| [o=k,break]));show&&print([u]);o} \\ optional args: show=1: print a(o..n-1), show=-1: append them on global list L, in both cases print [least unused number] at the end. See the wiki page for more.
%Y A329576 Cf. A329425 (6 primes using 5 consecutive terms), A329566 (6 primes using 6 consecutive terms).
%Y A329576 Cf. A329449 (4 primes using 4 consecutive terms), A329456 (4 primes using 5 consecutive terms).
%Y A329576 Cf. A329454 (3 primes using 4 consecutive terms), A329455 (3 primes using 5 consecutive terms).
%Y A329576 Cf. A329411 (2 primes using 3 consecutive terms), A329452 (2 primes using 4 consecutive terms), A329453 (2 primes using 5 consecutive terms).
%Y A329576 Cf. A329333 (1 (odd) prime using 3 terms), A128280 & A055265 (1 prime using 2 terms); A055266 & A253074 (0 primes using 2 terms), A329405 & A329450 (0 primes using 3 terms), A329406 - A329416, A329563 - A329581: other variants.
%K A329576 nonn
%O A329576 1,2
%A A329576 _M. F. Hasler_, Feb 09 2020