A294357 -id:A294357 - cp's OEIS frontend

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

R. Muller

nonn

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

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

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.

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

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

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

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 *)

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

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

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

Corrected and extended by David W. Wilson

A294294 Conjecturally, all odd numbers greater than a(n) can be represented in more ways by the sum of 3 odd primes p+q+r with p<=q<=r than a(n).

Original entry on oeis.org

7, 11, 15, 19, 23, 25, 31, 35, 37, 43, 45, 49, 55, 61, 63, 69, 75, 79, 81, 85, 87, 91, 99, 105, 111, 117, 129, 135, 141, 147, 159, 165, 171, 177, 195, 201, 207, 219, 225, 231, 237, 255, 261, 267, 279, 285, 291, 297, 309, 315, 321, 339, 345, 351

Offset: 1

Hugo Pfoertner, Oct 27 2017

nonn

The sequence provides numerical evidence of the validity of the ternary Goldbach conjecture, i.e. that every odd number >5 can be written as the sum of 3 primes, now proved by A. Helfgott.
The corresponding minimum numbers of representations are provided in A294295.
Empirically, mod(a(n),6) = 3 for all a(n) > 91 and mod(a(n),30) = 15 for all a(n) > 1281.

			a(1)=7 because all odd numbers > 7 have more representations by sums of 3 odd primes than 7, which has no such representation (A294295(1)=0).
a(2)=11, because all odd numbers > 11 have at least 2 representations p+q+r, e.g. 13=3+3+7=5+5+3 whereas 11=3+3+5 and 9=3+3+3 only have A294295(2)=1 representation.

For references and links see A007963.

Hugo Pfoertner, Table of n, a(n) for n = 1..301
H. A. Helfgott, The ternary Goldbach conjecture is true, arXiv:1312.7748 [math.NT], 2013-2014.

Cf. A007963, A102605, A139321, A294295, A294357, A294358.

A007963(k) > A007963((a(n)-1)/2) for all k > (a(n)-1)/2.

cp's OEIS Frontend

A294358 Number of ways to write A294357(n) as a sum of 3 odd primes p+q+r, with p>=q>=r.

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

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A294294 Conjecturally, all odd numbers greater than a(n) can be represented in more ways by the sum of 3 odd primes p+q+r with p<=q<=r than a(n).

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