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A329580 For every n >= 0, exactly 10 sums are prime among a(n+i) + a(n+j), 0 <= i < j < 8: lexicographically earliest such sequence of distinct nonnegative numbers.

%I A329580 #10 Nov 28 2019 11:03:18
%S A329580 0,1,2,3,4,5,6,90,7,11,8,9,10,12,13,30,29,31,14,16,15,17,22,42,19,25,
%T A329580 18,24,20,23,28,33,43,35,36,38,26,21,32,27,34,71,37,39,40,44,63,64,68,
%U A329580 41,46,183,50,45,333,51,98,47,58,62,69,65,48,101,66,49,61,78,57,53,180,52,55,96,631,54,56,83,75,95,74,116,60
%N A329580 For every n >= 0, exactly 10 sums are prime among a(n+i) + a(n+j), 0 <= i < j < 8: lexicographically earliest such sequence of distinct nonnegative numbers.
%C A329580 That is, there are 10 primes, counted with multiplicity, among the 28 pairwise sums of any 8 consecutive terms.
%C A329580 Is this a permutation of the nonnegative integers?
%C A329580 If so, then the restriction to [1..oo) is a permutation of the positive integers, but not the lexicographically earliest one with this property, which starts (1, 2, 3, 4, 5, 6, 7, 19, 10, 8, 9, 12, 11, 18, 13, 29, ...).
%C A329580 We remark the surprisingly large numbers 333 and 631 among the first terms.
%e A329580 In P(7) := {0, 1, 2, 3, 4, 5, 6} there are already S(7) := 10 primes 0+2, 0+3, 0+5, 1+2, 1+4, 1+6, 2+3, 2+5, 3+4, 5+6 among the pairwise sums, so the next term a(7) must not produce any more primes when added to elements of P(7). We find that a(7) = 90 is the smallest possible term.
%e A329580 Then in P(8) = {1, 2, 3, 4, 5, 6, 90} there are S(8) = 7 primes among the pairwise sums, so a(8) must produce 3 more primes when added to elements of P(8). We find a(8) = 7 is the smallest possibility (with 4+7, 6+7 and 90+7).
%e A329580 And so on.
%o A329580 (PARI) A329580(n,show=0,o=0,N=10,M=7,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 A329580 Cf. A329579 (9 primes using 7 consecutive terms), A329425 (6 primes using 5 consecutive terms).
%Y A329580 Cf. 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 A329580 Cf. A329577 (7 primes using 7 consecutive terms), A329566 (6 primes using 6 consecutive terms), A329449 (4 primes using 4 consecutive terms).
%Y A329580 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 A329580 Cf. A329405 ff: other variants defined for positive integers.
%K A329580 nonn
%O A329580 0,3
%A A329580 _M. F. Hasler_, Nov 17 2019