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A303603 a(n) is the maximum distance between primes in Goldbach partitions of 2n, or 2n if there are no Goldbach partitions of 2n. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 25 16:39 EDT 2021. Contains 346291 sequences. (Running on oeis4.)