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A054860
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Number of ways of writing 2n+1 as p + q + r where p, q, r are primes with p <= q <= r.
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7
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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
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OFFSET
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0,5
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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