A303603 a(n) is the maximum distance between primes in Goldbach partitions of 2n, or 2n if there are no Goldbach partitions of 2n.

0, 0, 0, 2, 4, 2, 8, 10, 8, 14, 16, 14, 20, 18, 16, 26, 28, 26, 24, 34, 32, 38, 40, 38, 44, 42, 40, 50, 48, 46, 56, 58, 56, 54, 64, 62, 68, 70, 68, 66, 76, 74, 80, 78, 76, 86, 84, 82, 60, 94, 92, 98, 100, 98, 104, 106, 104, 110, 108, 106, 96, 102, 100, 90, 124, 122, 128, 126, 124, 134, 136, 134, 132

Offset

1,4

Comments

The Goldbach Strong Conjecture is true if and only if a(n) = 2n for some n.

Terms are always even numbers because primes present in Goldbach partitions of n > 4 are odd and n = 4 has just one partition (2+2) where difference is 0.

Conjecture: Only first terms are 0 and all further terms are bigger than 0. Excluding a(1), a(n) = 0 iff the only Goldbach partition of 2n is n+n.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Marcin Barylski, C++ program

Marcin Barylski, Maximum distance for even numbers < 10^6

Eric Weisstein's MathWorld, Goldbach Partition

Formula

a(n) = 2 * A047949(n) if A047949(n) > 0 for n >= 2; a(n) = 2n if A047949(n) = -1. - Alois P. Heinz, Jun 01 2020

Example

a(1) = 0 for coherence with other related sequences.
a(2) = 0 because 2 * 2 = 4 = 2 + 2 and max_diff = 2 - 2 = 0.
a(8) = 10 because 2 * 8 = 16 = 5 + 11 = 3 + 13 and max_diff = 13 - 3 = 10.

Mathematica

a[1]=a[2]=0;
a[n_]:=Module[{p=3}, While[PrimeQ[2*n-p]!=True, p=NextPrime[p]]; 2*(n-p)];
a/@Range[73] (* Ivan N. Ianakiev, Jun 27 2018 *)

Prog

(PARI) a(n) = if (n==1, 0, forprime(p=2, , if (isprime(2*n-p), return (2*n-2*p)))); \\ Michel Marcus, Jul 02 2018

Crossrefs

Cf. A002372, A002375, A047949, A066285 (minimum distance), A305883.

Sequence in context: A296429 A065286 A068217 * A308044 A319252 A114593

Adjacent sequences: A303600 A303601 A303602 * A303604 A303605 A303606

Keyword

nonn

Author

Marcin Barylski, Apr 26 2018

Status

approved