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A007963 Number of (unordered) ways of writing 2n+1 as a sum of 3 odd primes. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

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

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

EXAMPLE

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

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 *)

PROG

(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

(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

(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

CROSSREFS

Cf. A068307, A087916.

Sequence in context: A092982 A248868 A030566 * A137222 A077641 A194210

Adjacent sequences:  A007960 A007961 A007962 * A007964 A007965 A007966

KEYWORD

nonn

AUTHOR

R. Muller

EXTENSIONS

Corrected and extended by David W. Wilson

STATUS

approved

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Last modified August 18 16:08 EDT 2017. Contains 290727 sequences.