A288574 -id:A288574 - cp's OEIS frontend

A288573 Erroneous version of A288574.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 4, 4, 6, 7, 9, 10, 11, 15, 17, 16, 19, 19, 23, 25, 26, 26, 28, 33, 32, 35, 43, 39, 40, 43, 43

Offset: 0

dead

V. Seleacu and I. Balacenoiu, 'Smarandache Notations (Book Series)', vol 10, American Research Press, 1999, p. 190.

K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996, 50 pages. See page 20.
K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996, 50 pages. [Cached copy] See page 20.

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

Original entry on oeis.org

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

James Sellers, May 25 2000

nonn

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

			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.

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.

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.

Cf. A007963, A288574.

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 *)

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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A288573 Erroneous version of A288574.

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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.

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