%I #7 Nov 28 2019 11:03:00

%S 0,1,2,3,4,5,20,9,10,8,33,11,6,50,21,17,56,12,47,14,26,7,125,15,24,83,

%T 54,66,13,35,22,18,19,48,23,31,28,30,25,16,36,42,121,29,43,37,46,70,

%U 72,60,27,79,67,40,34,39,32,69,38,41,44,45,51,58,62,86,52,53,105,171,65,74,146,68,63,123,76

%N For every n >= 0, exactly nine sums are prime among a(n+i) + a(n+j), 0 <= i < j < 7; lexicographically earliest such sequence of distinct nonnegative numbers.

%C That is, there are 9 primes, counted with multiplicity, among the 21 pairwise sums of any 7 consecutive terms.

%C Is this a permutation of the nonnegative integers?

%C If so, then the restriction to [1..oo) is a permutation of the positive integers, but maybe not the lexicographically earliest one with this property.

%o (PARI) A329579(n,show=0,o=0,N=9,M=6,p=[],U,u=o)={for(n=o,n-1, if(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||[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; o=1: start at a(1)=1; N, M: find N primes using M+1 terms

%Y Cf. A329577 (7 primes using 7 consecutive terms), A329566 (6 primes using 6 consecutive terms), A329449 (4 primes using 4 consecutive terms).

%Y Cf. A329425 (6 primes using 5 consecutive terms), A329455 (4 primes using 5 consecutive terms), A329455 (3 primes using 5 consecutive terms), A329453 (2 primes using 5 consecutive terms), A329452 (2 primes using 4 consecutive terms).

%Y Cf. A329454 (3 primes using 4 consecutive terms), A329411 (2 primes using 3 consecutive terms), A329333 (1 odd prime using 3 terms), A329450 (0 primes using 3 terms).

%Y Cf. A329405 ff: other variants defined for positive integers.

%K nonn

%O 0,3

%A _M. F. Hasler_, Nov 17 2019