OFFSET
1,5
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.
LINKS
T. D. Noe, Table of n, a(n) for n=1..10000
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
EXAMPLE
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.
MAPLE
for i from 1 to 500 do a[i] := 0:j := 1:while(ithprime(j)<i) do if(isprime(i-ithprime(j))=true) then a[i] := a[i]+1:fi:j := j+1:od:od:seq(a[k], k=1..500);
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 *)
PROG
(PARI) Vec(sum(i=1, 100, x^prime(i), O(x^prime(101)))^2) \\ Charles R Greathouse IV, Jan 21 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Aug 05 2002
EXTENSIONS
Corrected and extended by Vladeta Jovovic and Sascha Kurz, Aug 06 2002
STATUS
approved