A325142 a(n) = k if (n - k, n + k) is the centered Goldbach partition of 2n if it exists and -1 otherwise.

-1, -1, 0, 0, 1, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 9, 0, 5, 6, 3, 4, 9, 0, 1, 0, 9, 4, 3, 6, 5, 0, 9, 2, 3, 0, 1, 0, 3, 2, 15, 0, 5, 12, 3, 8, 9, 0, 7, 12, 3, 4, 15, 0, 1, 0, 9, 4, 3, 6, 5, 0, 15, 2, 3, 0, 1, 0, 15, 4, 3, 6, 5, 0, 9, 2, 15, 0

Offset: 0

Author

Peter Luschny, May 02 2019

Keywords

• sign

Comments

Let N = 2*n = p + q where p and q are primes. We call such a pair (p, q) a Goldbach partition of N. A centered Goldbach partition is the Goldbach partition of the form (n - k, n + k) where k >= 0 is minimal. If N has a centered Goldbach partition then a(n) is this k and otherwise -1.

According to Goldbach's conjecture, any even N = 2n > 2 has a Goldbach partition, which is necessarily of the form p = n - k, q = n + k: namely, with n = (p+q)/2 and k = (q-p)/2. - M. F. Hasler, May 02 2019

Examples

			a(162571) = 78 because 325142 = 162493 + 162649 and there is no k, 0 <= k < 78, such that (162571 - k, 162571 + k) is a Goldbach partition of 325142.
		

Links

• Peter Luschny, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A045917, A047160, A066285.

Cf. A002374, A112823, A002374, A234345, A020482.

Programs

Maple

a := proc(n) local k; for k from 0 to n do
if isprime(n + k) and isprime(n - k) then return k fi od: -1 end:
seq(a(n), n=0..83);

Mathematica

a[n_] := Module[{k}, For[k = 0, k <= n, k++, If[PrimeQ[n+k] && PrimeQ[n-k], Return[k]]]; -1]; Table[a[n], {n, 0, 83}] (* Jean-François Alcover, Jul 06 2019, from Maple *)

PARI

a(n) = for(k=0, n, if(ispseudoprime(n+k) && ispseudoprime(n-k), return(k))); -1 \\ Felix Fröhlich, May 02 2019

PARI

apply( A325142(n)=-!forprime(p=n,2*n, isprime(n*2-p)&&return(p-n)), [0..99]) \\ M. F. Hasler, May 02 2019

Formula

a(n) = n - A112823(n) = A234345(n) - n (= n - A002374(n) for n > 2). - M. F. Hasler, May 02 2019

a(n) = A047160(n) = A066285(n)/2 for n >= 2. - Alois P. Heinz, Jun 01 2020