A103151 Number of decompositions of 2n+1 into 2p+q, where p and q are both odd primes (A065091).

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, 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, 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

Offset: 1

Author

Lei Zhou, Feb 09 2005

Keywords

• nonn

Comments

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

Examples

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

Links

• R. J. Mathar, Table of n, a(n) for n = 1..10777

Crossrefs

A001031, A103152.

Programs

Maple

A103151 := proc(n)
    local s,a,q;
    a := 0 ;
    s := 2*n+1 ;
    for pi from 2 do
        q := s-2*ithprime(pi) ;
        if q <=2 then
            return a ;
        else
            if isprime(q) then
                a := a+1 ;
            end if;
        end if;
    end do:
end proc: # R. J. Mathar, Feb 22 2014

Mathematica

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}]

Extensions

Edited and Scheme-code added by Antti Karttunen, Jun 19 2007