A329425 For all n >= 0, six among (a(n+i) + a(n+j), 0 <= i < j < 5) are prime: lexicographically first such sequence of distinct nonnegative integers.

0, 1, 2, 3, 4, 9, 8, 10, 33, 14, 93, 20, 17, 23, 44, 6, 24, 35, 65, 5, 18, 32, 11, 12, 29, 30, 7, 31, 72, 16, 22, 25, 37, 15, 46, 64, 43, 28, 85, 19, 54, 13, 88, 34, 49, 39, 40, 27, 100, 57, 26, 52, 111, 21, 38, 45, 62, 41, 51, 56, 47, 116, 50, 81, 63, 68, 59, 170, 69, 71

Offset: 0

Author

M. F. Hasler, following an idea from Eric Angelini, Nov 24 2019

Keywords

• nonn

Comments

The restriction to [1, oo) is the lexicographically first such sequence of positive integers. (This is rather exceptional, cf. A128280 vs A055265, A329405 vs A329450, ..., see the wiki page for more.)

Conjectured to be a permutation, i.e., all n >= 0 appear. The restriction to [1, oo) is then the lexicographically first such permutation of the positive integers.

Among pairwise sums of 5 consecutive terms, there cannot be more than 2 x 3 = 6 primes: see the wiki page for this and further considerations and variants.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Éric Angelini, Prime sums from neighbouring terms, SeqFan list, and personal blog "Cinquante signes", 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. A055265, A128280 (1 prime from 2 terms), A329333 (1 prime from 3 terms), A329405-A329416 (N primes from M terms >= 1), A329449, ..., A329581 (N primes from M terms >= 0).

Programs

Maple

R:= 0,1,2,3,4:
S:= {R}:
for i from 1 to 100 do
  for x from 5 do
    if member(x,S) then next fi;
    n1:= nops(select(isprime,[seq(seq(R[i+j]+R[i+k],j=1..k-1),k=1..4)]));
    if nops(select(isprime,[seq(R[i+j]+x,j=1..4)]))+n1 = 6 then
      R:= R, x; S:= S union {x}; break
    fi
od od:
R; # Robert Israel, Dec 29 2022

PARI

A329425_upto(N) = S(N,6,5,0) \\ see the wiki page for the function S().