

A068307


From Goldbach problem: number of decompositions of n into a sum of three primes.


40



0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 4, 2, 3, 2, 5, 2, 5, 3, 5, 3, 7, 3, 7, 2, 6, 3, 9, 2, 8, 4, 9, 4, 10, 2, 11, 3, 10, 4, 12, 3, 13, 4, 12, 5, 15, 4, 16, 3, 14, 5, 17, 3, 16, 4, 16, 6, 19, 3, 21, 5, 20, 6, 20, 2, 22, 5, 21, 6, 22, 5, 28, 5, 24, 7
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,9


COMMENTS

For even n>2, a(n) = A061358(n2).  Reinhard Zumkeller, Aug 08 2009
Vinogradov proved that every sufficiently large odd number is the sum of three primes.  T. D. Noe, Mar 27 2013
The two Helfgott papers show that every odd number greater than 5 is the sum of three primes (this is the Odd Goldbach Conjecture).  T. D. Noe, May 14 2013, N. J. A. Sloane, May 18 2013


LINKS

T. D. Noe and Robert G. Wilson v, Table of n, a(n) for n=1..36000 (first 10000 terms from T. D. Noe)
H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252 [math.NT], 2012.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 2013.
H. A. Helfgott, The ternary Goldbach conjecture is true, arxiv:1312.7748 [math.NT], 2013.
H. A. Helfgott, The ternary Goldbach problem, arXiv:1404.2224 [math.NT], 2014.
Yannick Saouter, Checking the odd Goldbach conjecture up to 10^20, Math. Comp. 67 (222) (1998) 863866.
Eric W. Weinstein, MathWorld: Vinogradov's Theorem
Wikipedia, Goldbach's conjecture.


EXAMPLE

a(6) = 1 because 6 = 2+2+2,
a(9) = 2 because 9 = 2+2+5 = 3+3+3,
a(15) = 3 because 15 = 2+2+11 = 3+5+7 = 5+5+5,
a(17) = 4 because 17 = 2+2+13 = 3+3+11 = 3+7+7 = 5+5+7.
 Zak Seidov, Jun 29 2017


MATHEMATICA

f[n_] := Block[{c = 0, lmt = PrimePi@ Floor[n/2], p, q}, Do[p = Prime@ i; q = Prime@ j; r = n  p  q; If[ PrimeQ@ r && r >= p, c++ ], {i, lmt}, {j, i}]; c]; Array[f, 91] (* Robert G. Wilson v, Apr 13 2008 *)


PROG

(PARI) a(n)=my(s); forprime(p=(n+2)\3, n4, forprime(q=(np+1)\2, min(np2, p), if(isprime(npq), s++))); s \\ Charles R Greathouse IV, Jun 29 2017
(Python)
from sympy import isprime, primerange, floor
def a(n):
s=0
for p in primerange(floor((n + 2)/3), n  3):
for q in primerange(floor((n  p + 1)/2), min(n  p  2, p) + 1):
if isprime(n  p  q): s+=1
return s
print [a(n) for n in xrange(1, 101)] # Indranil Ghosh, Jul 01 2017, after PARI code by Charles R Greathouse IV


CROSSREFS

Bisections: A045917, A054860. Cf. A002375, A007963, A061358, A059998.
First occurrence: A139321. Records: A139322.
Column k=3 of A117278.
Sequence in context: A001227 A060764 A105149 * A158946 A223853 A023645
Adjacent sequences: A068304 A068305 A068306 * A068308 A068309 A068310


KEYWORD

easy,nonn


AUTHOR

Naohiro Nomoto, Feb 24 2002


EXTENSIONS

More terms from Vladeta Jovovic, Mar 10 2002


STATUS

approved



