%I #8 Feb 22 2014 17:55:30

%S 0,0,0,1,1,2,1,3,2,2,2,3,3,4,2,4,2,4,4,4,4,5,3,4,6,5,3,6,3,3,6,6,5,7,

%T 3,4,7,6,5,8,3,7,7,7,4,10,5,6,9,5,5,11,5,6,9,7,6,10,7,5,11,8,6,10,5,6,

%U 12,8,5,12,5,9,12,8,6,13,7,6,11,9,9,16,4,8,12,9,9,13,7,6,13,11,8,16,6

%N Number of decompositions of 2n+1 into 2p+q, where p and q are both odd primes (A065091).

%C Conjecture: all items for n>=4 are greater than or equal to 1. This is a stronger conjecture than the Goldbach conjecture.

%H R. J. Mathar, <a href="/A103151/b103151.txt">Table of n, a(n) for n = 1..10777</a>

%e For 2*4+1 = 9 we have just one such composition: 9 = 2*3+3, so a(4)=1;

%e For 2*14+1 = 29 we have four such compositions: 29 = 2*3+23 = 2*5+19 = 2*11+7 = 2*13+3, so a(14)=4.

%p A103151 := proc(n)

%p local s,a,q;

%p a := 0 ;

%p s := 2*n+1 ;

%p for pi from 2 do

%p q := s-2*ithprime(pi) ;

%p if q <=2 then

%p return a ;

%p else

%p if isprime(q) then

%p a := a+1 ;

%p end if;

%p end if;

%p end do:

%p end proc: # _R. J. Mathar_, Feb 22 2014

%t Do[m = 3; ct = 0; While[(m*2) < n, If[PrimeQ[m], cp = n - (2*m); If[ PrimeQ[cp], ct = ct + 1]]; m = m + 2]; Print[ct], {n, 9, 299, 2}]

%o (Scheme, with Aubrey Jaffer's SLIB Scheme library from http://www.swiss.ai.mit.edu/~jaffer/SLIB.html )

%o (define (A103151 n) (let loop ((i 2) (z 0)) (let ((p1 (A000040 i))) (cond ((>= p1 n) z) ((prime? (+ 1 (* 2 (- n p1)))) (loop (+ 1 i) (+ 1 z))) (else (loop (+ 1 i) z))))))

%Y A001031, A103152.

%K nonn

%O 1,6

%A _Lei Zhou_, Feb 09 2005

%E Edited and Scheme-code added by _Antti Karttunen_, Jun 19 2007