A294294 -id:A294294 - 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

A294357 Smallest odd number that can be expressed in more ways by sums of 3 odd primes p+q+r with p <= q <= r than any smaller odd number.

Original entry on oeis.org

9, 13, 17, 21, 25, 27, 29, 33, 37, 39, 45, 47, 51, 53, 63, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 161, 167, 173, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 247, 251, 257, 269, 277, 281, 287, 293, 299

Offset: 1

Hugo Pfoertner, Oct 29 2017

nonn

Position of n-th record in A007963 converted to actual odd number for which the record is achieved.
The corresponding records of numbers of representations are provided in A294358.
Empirically mod(a(n),6) = 5 for all a(n) > 63 and mod(a(n),30) != 5 for all a(n) > 425.

Hugo Pfoertner, Table of n, a(n) for n = 1..1113

Cf. A007963, A102605, A139321, A294294, A294295, A294358.

a(1)=9 because 9 = 3+3+3 is the smallest number that can be represented as sum of 3 odd primes.

a(13)=51 because A007963(25) = A007963((51-1)/2) = 14 is the 13th record in A007963.

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A294295 Number of ways to write A294294(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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A294357 Smallest odd number that can be expressed in more ways by sums of 3 odd primes p+q+r with p <= q <= r than any smaller odd number.

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