A073610 Number of primes of the form n-p where p is a prime.

0, 0, 0, 1, 2, 1, 2, 2, 2, 3, 0, 2, 2, 3, 2, 4, 0, 4, 2, 4, 2, 5, 0, 6, 2, 5, 0, 4, 0, 6, 2, 4, 2, 7, 0, 8, 0, 3, 2, 6, 0, 8, 2, 6, 2, 7, 0, 10, 2, 8, 0, 6, 0, 10, 2, 6, 0, 7, 0, 12, 2, 5, 2, 10, 0, 12, 0, 4, 2, 10, 0, 12, 2, 9, 2, 10, 0, 14, 0, 8, 2, 9, 0, 16, 2, 9, 0, 8, 0, 18, 2, 8, 0, 9, 0, 14, 0, 6

Offset: 1

Author

Amarnath Murthy, Aug 05 2002

Keywords

• nonn

Comments

a(p) = 2 if p-2 is a prime else a(p) = 0. If n = 2p, p is a prime then a(n) is odd else a(n) is even. As p is counted only once and if q and n-q both are prime then the count is increased by 2. ( Analogous to the fact that perfect squares have odd number of divisors).

a(2k+1) = 2 if (2k-1) is prime, else a(2k+1)=0 (for any k). This sequence can be used to re-describe a couple of conjectures: the Goldbach conjecture == a(2n) > 0 for all n>=2; twin primes conjecture == for any n, there is a prime p>n s.t. a(p)>0.

Number of ordered ways of writing n as the sum of two primes.

Examples

			a(16) = 4 as there are 4 primes 3,5,11 and 13 such that 16-3,16-5,16-11and 16-13 are primes.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..10000

Crossrefs

Cf. A061358, A065577, A107318, A098238

Programs

Maple

for i from 1 to 500 do a[i] := 0:j := 1:while(ithprime(j)
				

Mathematica

nn=20;a[x]:=Sum[x^i,{i,Table[Prime[n],{n,1,nn}]}];Drop[CoefficientList[a[x]^2,x],1]  (* Geoffrey Critzer, Nov 22 2012 *)

PARI

Vec(sum(i=1,100,x^prime(i),O(x^prime(101)))^2) \\ Charles R Greathouse IV, Jan 21 2015

Formula

G.f.: (Sum_{k>0} x^prime(k))^2. - Vladeta Jovovic, Mar 12 2005

Self-convolution of characteristic function of primes (A010051). - Graeme McRae, Jul 18 2006

Extensions

Corrected and extended by Vladeta Jovovic and Sascha Kurz, Aug 06 2002