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 A054860 Number of ways of writing 2n+1 as p + q + r where p, q, r are primes with p <= q <= r. 7
 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, 16, 16, 19, 21, 20, 20, 22, 21, 22, 28, 24, 25, 29, 27, 29, 33, 29, 33, 35, 34, 30, 38, 36, 35, 43, 38, 37, 47, 42, 43, 50, 46, 47, 53, 50, 45, 57, 54, 47, 62, 53, 49, 65, 59, 55, 68 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. The unconditional theorem was proved by Helfgott (see link below). - T. D. Noe, May 15 2013 REFERENCES G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, appendix 3. Wolfgang Schwarz, Einfuehrung in Methoden und Ergebnisse der Primzahltheorie, Bibliographisches Institut Mannheim, 1969, ch. 7. LINKS Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (up to 1000 from T. D. Noe) H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv 1305.2897 [math.NT], May 14 2013. EXAMPLE 7 = 2 + 2 + 3 so a(3) = 1; 9 = 2 + 2 + 5 = 3 + 3 + 3 so a(4) = 2; 11 = 2 + 2 + 7 = 3 + 3 + 5 so a(5) = 2. MATHEMATICA nn = 201; t = Table[0, {(nn + 1)/2}]; pMax = PrimePi[nn]; ps = Prime[Range[pMax]]; Do[n = ps[[i]] + ps[[j]] + ps[[k]]; If[n <= nn && OddQ[n], t[[(n + 1)/2]]++], {i, pMax}, {j, i, pMax}, {k, j, pMax}]; t (* T. D. Noe, May 23 2017 *) f[n_] := Length@ IntegerPartitions[2n +1, {3}, Prime@ Range@ PrimePi[2n -3]]; Array[f, 75, 0] (* Robert G. Wilson v, Jun 30 2017 *) PROG (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 CROSSREFS Cf. A007963, A288574. Sequence in context: A241952 A236919 A138304 * A098745 A029158 A241953 Adjacent sequences:  A054857 A054858 A054859 * A054861 A054862 A054863 KEYWORD nonn AUTHOR James A. Sellers, May 25 2000 STATUS approved

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Last modified December 13 20:27 EST 2019. Contains 329973 sequences. (Running on oeis4.)