cp's OEIS Frontend

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

%I A073610 #25 May 10 2020 15:49:31
%S A073610 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,
%T A073610 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,
%U A073610 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
%N A073610 Number of primes of the form n-p where p is a prime.
%C A073610 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).
%C A073610 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.
%C A073610 Number of ordered ways of writing n as the sum of two primes.
%H A073610 T. D. Noe, <a href="/A073610/b073610.txt">Table of n, a(n) for n=1..10000</a>
%F A073610 G.f.: (Sum_{k>0} x^prime(k))^2. - _Vladeta Jovovic_, Mar 12 2005
%F A073610 Self-convolution of characteristic function of primes (A010051). - _Graeme McRae_, Jul 18 2006
%e A073610 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.
%p A073610 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);
%t A073610 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 *)
%o A073610 (PARI) Vec(sum(i=1,100,x^prime(i),O(x^prime(101)))^2) \\ _Charles R Greathouse IV_, Jan 21 2015
%Y A073610 Cf. A061358, A065577, A107318, A098238
%K A073610 nonn
%O A073610 1,5
%A A073610 _Amarnath Murthy_, Aug 05 2002
%E A073610 Corrected and extended by _Vladeta Jovovic_ and _Sascha Kurz_, Aug 06 2002