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%I #35 Jun 30 2017 17:26:49
%S 0,0,0,1,2,2,2,3,4,3,5,5,5,7,7,6,9,8,9,10,11,10,12,13,12,15,16,14,17,
%T 16,16,19,21,20,20,22,21,22,28,24,25,29,27,29,33,29,33,35,34,30,38,36,
%U 35,43,38,37,47,42,43,50,46,47,53,50,45,57,54,47,62,53,49,65,59,55,68
%N Number of ways of writing 2n+1 as p + q + r where p, q, r are primes with p <= q <= r.
%C Every sufficiently large odd number is the sum of three primes (th. by Vinogradov, 1937). Goldbach's conjecture requires three ODD primes and then a(n) > 0 for n > 2 is weaker.
%C The unconditional theorem was proved by Helfgott (see link below). - _T. D. Noe_, May 15 2013
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, appendix 3.
%D Wolfgang Schwarz, Einfuehrung in Methoden und Ergebnisse der Primzahltheorie, Bibliographisches Institut Mannheim, 1969, ch. 7.
%H Charles R Greathouse IV, <a href="/A054860/b054860.txt">Table of n, a(n) for n = 0..10000</a> (up to 1000 from _T. D. Noe_)
%H H. A. Helfgott, <a href="http://arxiv.org/abs/1305.2897">Major arcs for Goldbach's theorem</a>, arXiv 1305.2897 [math.NT], May 14 2013.
%e 7 = 2 + 2 + 3 so a(3) = 1;
%e 9 = 2 + 2 + 5 = 3 + 3 + 3 so a(4) = 2;
%e 11 = 2 + 2 + 7 = 3 + 3 + 5 so a(5) = 2.
%t nn = 201; t = Table[0, {(nn + 1)/2}]; pMax = PrimePi[nn]; ps =
%t Prime[Range[pMax]]; Do[n = ps[[i]] + ps[[j]] + ps[[k]]; If[n <= nn &&
%t OddQ[n], t[[(n + 1)/2]]++], {i, pMax}, {j, i, pMax}, {k, j, pMax}]; t (* _T. D. Noe_, May 23 2017 *)
%t f[n_] := Length@ IntegerPartitions[2n +1, {3}, Prime@ Range@ PrimePi[2n -3]]; Array[f, 75, 0] (* _Robert G. Wilson v_, Jun 30 2017 *)
%o (PARI) first(n)=my(v=vector(n)); forprime(r=3,2*n-3, v[r\2+2]++); forprime(p=3,(2*n+1)\3, forprime(q=p,(2*n+1-p)\2, forprime(r=q,2*n+1-p-q, v[(p+q+r)\2]++))); concat(0, v) \\ _Charles R Greathouse IV_, May 25 2017
%Y Cf. A007963, A288574.
%K nonn
%O 0,5
%A _James A. Sellers_, May 25 2000
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