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A329568 For all n >= 1, exactly 9 sums are prime among a(n+i) + a(n+j), 0 <= i < j < 6: lexicographically earliest such sequence of distinct positive numbers.

%I A329568 #25 Dec 23 2024 14:53:46
%S A329568 1,2,3,9,4,10,27,14,33,57,26,40,87,50,21,63,16,20,51,8,81,93,46,56,15,
%T A329568 58,135,183,28,44,39,88,69,123,34,68,105,128,45,129,22,52,141,38,75,
%U A329568 159,32,82,99,64,117,147,80,94,177,116,237,273,74,100,387,76,207,357,62,104,165,86,77,95
%N A329568 For all n >= 1, exactly 9 sums are prime among a(n+i) + a(n+j), 0 <= i < j < 6: lexicographically earliest such sequence of distinct positive numbers.
%C A329568 That is, there are nine primes, counted with multiplicity, among the 15 pairwise sums of any six consecutive terms. This is the maximum number of possible prime sums for any set of 6 numbers > 1, see wiki page for details.
%C A329568 Conjectured to be a permutation of the positive integers. See A329569 = (0, 1, 2, 5, 6, 11, 12, 17, ...) for the quite different variant for nonnegative integers.
%C A329568 For n > 6, a(n) is the smallest number not used earlier such that the set a(n) + {a(n-5), ..., a(n-1)} has the same number of primes as a(n-6) + {a(n-5), ..., a(n-1)}. Such a number always exists, by definition of the sequence. (If it would not exist for a given n, the term a(n-1) (or earlier) "was wrong and must be corrected", so to say.) See the wiki page for further considerations about existence and surjectivity.
%C A329568 For a(4), one must exclude the values {4, ..., 8} to get an infinite sequence, but for all other (at least several hundred) terms, the greedy choice gives the correct solution.
%H A329568 Éric Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2019-November/020145.html">Prime sums from neighbouring terms</a>, SeqFan list, Nov 11 2019.
%H A329568 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.
%o A329568 (PARI) {A329568(n,show=0,o=1,N=9,M=5,X=[[4,x]|x<-[4..8]],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])))); for(k=u,oo,bittest(U,k-u)|| min(c-#[0|x<-p,isprime(x+k)],#p>=M)|| 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. Parameters N,M,o,... allow getting other variants, see the wiki page for more.
%Y A329568 Cf. A055265, A128280 (1 prime from 2 terms), A329333 (1 prime from 3 terms), A329405, ..., A329417 (N primes from M terms >= 1), A329425, A329449, ..., A329581 (N primes from M terms >= 0).
%K A329568 nonn
%O A329568 1,2
%A A329568 _M. F. Hasler_, Feb 10 2020