A329452 There are exactly two primes in {a(n+i) + a(n+j), 0 <= i < j <= 3} for any n: lexicographically earliest such sequence of distinct nonnegative integers.

0, 1, 2, 8, 4, 5, 6, 3, 7, 11, 10, 9, 12, 13, 28, 15, 17, 16, 20, 14, 21, 22, 19, 23, 25, 24, 29, 30, 26, 18, 35, 31, 32, 27, 34, 36, 33, 38, 37, 40, 63, 39, 41, 44, 42, 45, 47, 50, 51, 43, 52, 49, 46, 48, 53, 54, 57, 55, 56, 58, 69, 62, 59, 65, 66, 61, 60, 67, 64, 68, 70, 81, 72, 76, 73, 75, 71

Offset: 0

Author

M. F. Hasler, Nov 15 2019

Keywords

• nonn

Comments

That is, there are exactly two primes among the 6 pairwise sums of any four consecutive terms.

Conjectured to be a permutation of the nonnegative numbers.

a(100) = 97, a(1000) = 1001, a(10^4) = 9997, a(10^5) = 10^5, a(10^6) = 999984 and all numbers below 999963 have appeared at that point.

See the wiki page for considerations about existence and surjectivity of the sequence and variants thereof.

Examples

			We start with a(0) = 0, a(1) = 1, a(2) = 2, the smallest possibilities which do not lead to a contradiction.
Now there are already 2 primes, 0 + 2 and 1 + 2, among the pairwise sums, so the next term must not generate any further prime. Given 0 and 1, primes and (primes - 1) are excluded, and a(3) = 8 is the smallest possible choice.
Now there is only one prime, 1 + 2 = 3, among the pairwise sums using {1, 2, 8}; the next term must produce exactly one additional prime as sum with these. We see that 3 is not possible (2 + 3 = 5 and 8 + 3 = 11), but a(4) = 4 is possible.
Now using {2, 8, 4} we have no prime as a pairwise sum, so the next term must produce two primes among the sums with these terms. Again, 3 would give three primes, but 5 yields exactly two primes, 2 + 5 = 7 and 8 + 5 = 13.
		

Links

• Eric Angelini, Prime sums from neighbouring terms, personal blog "Cinquante signes" (and post to the SeqFan list), Nov. 11, 2019.
• Eric Angelini, Prime sums from neighbouring terms [Cached copy of html file, with permission]
• Eric Angelini, Prime sums from neighbouring terms [Cached copy of pdf file, with permission]
• M. F. Hasler, Prime sums from neighboring terms, OEIS wiki, Nov. 23, 2019

Crossrefs

Cf. A329412 (analog for positive integers), A329453 (2 primes in a(n+i)+a(n+j), i < j < 5).

Cf. A329333 (one odd prime among a(n+i)+a(n+j), 0 <= i < j < 3), A329450 (no prime in a(n+i)+a(n+j), i < j < 3).

Programs

PARI

A329452(n,show=0,o=0,p=[],U,u=o)={for(n=o,n-1, show&&print1(o","); U+=1<<(o-u); U>>=-u+u+=valuation(U+1,2); p=concat(if(2<#p,p[^1],p),o); my(c=2-sum(i=2,#p,sum(j=1,i-1,isprime(p[i]+p[j])))); if(#p<3, o=u;next); for(k=u,oo, bittest(U,k-u) || sum(i=1,#p,isprime(p[i]+k))!=c || [o=k, break]));print([u]);o} \\ Optional args: show=1: print a(o..n-1); o=1: use indices & terms >= 1, i.e., compute A329412. See the wiki page for more general code returning a vector: S(n,2,4) = a(0..n-1).

Extensions

Edited (deleted comments now found on the wiki) by M. F. Hasler, Nov 24 2019