A054860 Number of ways of writing 2n+1 as p + q + r where p, q, r are primes with p <= q <= r.

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

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