%I #18 Jan 02 2023 12:30:54

%S 0,1,2,5,6,11,12,17,26,35,36,47,24,77,32,65,62,149,74,9,8,39,14,15,4,

%T 3,28,33,38,69,10,51,20,21,58,93,16,81,46,13,70,27,76,37,34,97,52,7,

%U 30,49,40,31,22,67,82,19,42,25,64,85,18,109,54,43,88,139,84,145,94,79,112,55,48,289,144

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

%C That is, there are nine primes, counted with multiplicity, among the 15 pairwise sums of any six consecutive terms. This is the maximum: there can't be more than 9 primes among the pairwise sums of any 6 numbers > 1, cf. wiki page in LINKS.

%C Conjectured to be a permutation of the nonnegative integers. The restriction to [1,oo) is then a permutation of the positive integers with similar properties, but different from the lexico-smallest one, A329568 = (1, 2, 3, 9, 4, 10, 27, ...).

%C For n > 5, 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) "is wrong and must be corrected", so to say.) See the wiki page for further considerations about existence and surjectivity.

%C For a(3) and a(4), one must exclude values 3 & 4 to be able to continue the sequence indefinitely, but in all other cases (at least for several hundred terms), the greedy choice gives the correct solution.

%C The values 3, 4 and 7 appear quite late at indices 25, 24 resp. 47.

%H Éric Angelini, <a href="http://list.seqfan.eu/oldermail/seqfan/2019-November/020145.html">Prime sums from neighbouring terms</a>, SeqFan list, 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.

%o (PARI) {A329569(n,show=0,o=0,N=9,M=5,X=[[3,3],[3,4],[4,3],[4,4]],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 Cf. A055265, A128280 (1 prime from 2 terms), A329333 (1 prime from 3 terms), A329405, ..., A329416 (N primes from M terms >= 1), A329425, A329449, ..., A329581 (N primes from M terms >= 0).

%K nonn

%O 0,3

%A _M. F. Hasler_, Feb 10 2020