cp's OEIS Frontend

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

%I A303603 #72 Feb 16 2025 08:33:54
%S A303603 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,
%T A303603 42,40,50,48,46,56,58,56,54,64,62,68,70,68,66,76,74,80,78,76,86,84,82,
%U A303603 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
%N A303603 a(n) is the maximum distance between primes in Goldbach partitions of 2n, or 2n if there are no Goldbach partitions of 2n.
%C A303603 The Goldbach Strong Conjecture is true if and only if a(n) = 2n for some n.
%C A303603 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.
%C A303603 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.
%H A303603 Seiichi Manyama, <a href="/A303603/b303603.txt">Table of n, a(n) for n = 1..10000</a>
%H A303603 Marcin Barylski, <a href="/A303603/a303603_2.cpp.txt">C++ program</a>
%H A303603 Marcin Barylski, <a href="/A303603/a303603.png">Maximum distance for even numbers < 10^6</a>
%H A303603 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%F A303603 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
%e A303603 a(1) = 0 for coherence with other related sequences.
%e A303603 a(2) = 0 because 2 * 2 = 4 = 2 + 2 and max_diff = 2 - 2 = 0.
%e A303603 a(8) = 10 because 2 * 8 = 16 = 5 + 11 = 3 + 13 and max_diff = 13 - 3 = 10.
%t A303603 a[1]=a[2]=0;
%t A303603 a[n_]:=Module[{p=3},While[PrimeQ[2*n-p]!=True,p=NextPrime[p]];2*(n-p)];
%t A303603 a/@Range[73] (* _Ivan N. Ianakiev_, Jun 27 2018 *)
%o A303603 (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
%Y A303603 Cf. A002372, A002375, A047949, A066285 (minimum distance), A305883.
%K A303603 nonn
%O A303603 1,4
%A A303603 _Marcin Barylski_, Apr 26 2018