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%I
%S 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,
%T 16,16,18,20,20,20,21,21,21,27,24,25,28,27,28,33,29,32,35,34,30,37,36,
%U 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
%N Number of (unordered) ways of writing 2n+1 as a sum of 3 odd primes.
%C Ways of writing 2n+1 as p+q+r where p,q,r are odd primes with p <= q <= r.
%C 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
%D George E. Andrews, Number Theory (NY, Dover, 1994), page 111.
%D Ivars Peterson, The Mathematical Tourist (NY, W. H. Freeman, 1998, pages 35-37.
%D Paulo Ribenboim, "VI, Goldbach's famous conjecture," The New Book of Prime Number Records, 3rd ed. (NY, Springer, 1996), pages 291-299.
%H T. D. Noe, <a href="/A007963/b007963.txt">Table of n, a(n) for n = 0..10000</a>
%H H. A. Helfgott, <a href="http://arxiv.org/abs/1205.5252">Minor arcs for Goldbach's problem</a>, arXiv:1205.5252, 2012.
%H H. A. Helfgott, <a href="http://arxiv.org/abs/1305.2897">Major arcs for Goldbach's theorem</a>, arXiv:1305.2897.
%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>.
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e a(10) = 4 because 21 = 3+5+13 = 3+7+11 = 5+5+11 = 7+7+7.
%o (Sage)
%o def A007963(n):
%o c = 0
%o for p in Partitions(n, length = 3):
%o b = True
%o for t in p:
%o b = is_prime(t) and t > 2
%o if not b: break
%o if b : c = c + 1
%o return c
%o [A007963(2*n+1) for n in (0..77)] # _Peter Luschny_, May 18 2013
%Y Cf. A068307.
%K nonn,changed
%O 0,7
%A R. Muller
%E Corrected and extended by _David W. Wilson_
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