A007963 Number of (unordered) ways of writing 2n+1 as a sum of 3 odd primes.

0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 6, 8, 7, 9, 10, 10, 10, 11, 12, 12, 14, 16, 14, 16, 16, 16, 18, 20, 20, 20, 21, 21, 21, 27, 24, 25, 28, 27, 28, 33, 29, 32, 35, 34, 30, 37, 36, 34, 42, 38, 36, 46, 42, 42, 50, 46, 47, 53, 50, 45, 56, 54, 46, 62, 53, 48, 64, 59, 55, 68, 61, 59, 68

Offset: 0

Author

R. Muller

Keywords

• nonn

Comments

Ways of writing 2n+1 as p+q+r where p,q,r are odd primes with p <= q <= r.

The two papers of Helfgott appear to provide a proof of the Odd Goldbach Conjecture that every odd number greater than five is the sum of three primes. (The paper is still being reviewed.) - Peter Luschny, May 18 2013; N. J. A. Sloane, May 19 2013

Examples

			a(10) = 4 because 21 = 3+5+13 = 3+7+11 = 5+5+11 = 7+7+7.
		

References

• George E. Andrews, Number Theory (NY, Dover, 1994), page 111.
• Ivars Peterson, The Mathematical Tourist (NY, W. H. Freeman, 1998), pages 35-37.
• Paulo Ribenboim, "VI, Goldbach's famous conjecture," The New Book of Prime Number Records, 3rd ed. (NY, Springer, 1996), pages 291-299.

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252 [math.NT], 2012.
• H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 2013.
• H. A. Helfgott, The ternary Goldbach conjecture is true, arxiv:1312.7748 [math.NT], 2013.
• H. A. Helfgott, The ternary Goldbach problem, arXiv:1404.2224 [math.NT], 2014.
• F. Smarandache, Only Problems, Not Solutions!.
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A068307, A087916, A294294 (lower bound of scatterplot), A294357, A294358 (records).

Programs

Maple

A007963 := proc(n)
    local a,i,j,k,p,q,r ;
    a := 0 ;
    for i from 2 do
        p := ithprime(i) ;
        for j from i do
            q := ithprime(j) ;
            for k from j do
                r := ithprime(k) ;
                if p+q+r = 2*n+1 then
                    a := a+1 ;
                elif p+q+r > 2*n+1 then
                    break;
                end if;
            end do:
            if p+2*q > 2*n+1 then
                break;
            end if;
        end do:
        if 3*p > 2*n+1 then
            break;
        end if;
    end do:
    return a;
end proc:
seq(A007963(n),n=0..30) ; # R. J. Mathar, Sep 06 2014

Mathematica

nn = 75; ps = Prime[Range[2, nn + 1]]; c = Flatten[Table[If[i >= j >= k, i + j + k, 0], {i, ps}, {j, ps}, {k, ps}]]; Join[{0, 0, 0, 0}, Transpose[Take[Rest[Sort[Tally[c]]], nn+2]][[2]]] (* T. D. Noe, Apr 08 2014 *)

PARI

a(n)=my(k=2*n+1,s,t); forprime(p=(k+2)\3,k-6, t=k-p; forprime(q=t\2,min(t-3,p), if(isprime(t-q), s++))); s \\ Charles R Greathouse IV, Mar 20 2017

Perl

use ntheory ":all"; sub a007963 { my($n,$c)=(shift,0); forpart { $c++ if vecall { is_prime($) } @; } $n,{n=>3,amin=>3}; $c; }
say "$ ",a007963(2*$+1) for 0..100; # Dana Jacobsen, Mar 19 2017

Sage

def A007963(n):
    c = 0
    for p in Partitions(n, length = 3):
        b = True
        for t in p:
            b = is_prime(t) and t > 2
            if not b: break
        if b : c = c + 1
    return c
[A007963(2*n+1) for n in (0..77)]   # Peter Luschny, May 18 2013

Extensions

Corrected and extended by David W. Wilson