A068307 From Goldbach problem: number of decompositions of n into a sum of three primes.

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

Offset: 1

Author

Naohiro Nomoto, Feb 24 2002

Keywords

• easy
• nonn

Comments

For even n > 2, a(n) = A061358(n-2). - 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

Examples

			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
		

Links

• 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) 863-866.
• Eric Weisstein's World of Mathematics, Vinogradov's Theorem
• Wikipedia, Goldbach's conjecture.

Crossrefs

Bisections: A045917, A054860. Cf. A002375, A007963, A061358, A059998.

First occurrence: A139321. Records: A139322.

Column k=3 of A117278.

Programs

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 *)
Table[Count[IntegerPartitions[n,{3}],?(AllTrue[#,PrimeQ]&)],{n,50}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Sep 10 2019 *)

PARI

a(n)=my(s); forprime(p=(n+2)\3,n-4, forprime(q=(n-p+1)\2,min(n-p-2,p), if(isprime(n-p-q), 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(((n + 2)//3), n - 3):
        for q in primerange(((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 range(1, 101)]) # Indranil Ghosh, Jul 01 2017, after PARI code by Charles R Greathouse IV

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} A010051(i) * A010051(k) * A010051(n-i-k). - Wesley Ivan Hurt, Mar 26 2019

a(n) = [x^n y^3] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019

Extensions

More terms from Vladeta Jovovic, Mar 10 2002