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%I #78 Jul 11 2021 12:15:54
%S 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,
%T 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,
%U 19,3,21,5,20,6,20,2,22,5,21,6,22,5,28,5,24,7
%N From Goldbach problem: number of decompositions of n into a sum of three primes.
%C For even n > 2, a(n) = A061358(n-2). - _Reinhard Zumkeller_, Aug 08 2009
%C Vinogradov proved that every sufficiently large odd number is the sum of three primes. - _T. D. Noe_, Mar 27 2013
%C 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
%H Robert G. Wilson v, <a href="/A068307/b068307.txt">Table of n, a(n) for n = 1..36000</a> (first 10000 terms from T. D. Noe)
%H H. A. Helfgott, <a href="http://arxiv.org/abs/1205.5252">Minor arcs for Goldbach's problem</a>, arXiv:1205.5252 [math.NT], 2012.
%H H. A. Helfgott, <a href="http://arxiv.org/abs/1305.2897">Major arcs for Goldbach's theorem</a>, arXiv:1305.2897 [math.NT], 2013.
%H H. A. Helfgott, <a href="http://arxiv.org/abs/1312.7748">The ternary Goldbach conjecture is true</a>, arxiv:1312.7748 [math.NT], 2013.
%H H. A. Helfgott, <a href="http://arxiv.org/abs/1404.2224">The ternary Goldbach problem</a>, arXiv:1404.2224 [math.NT], 2014.
%H Yannick Saouter, <a href="http://dx.doi.org/10.1090/S0025-5718-98-00928-4">Checking the odd Goldbach conjecture up to 10^20</a>, Math. Comp. 67 (222) (1998) 863-866.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VinogradovsTheorem.html">Vinogradov's Theorem</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach's_conjecture">Goldbach's conjecture</a>.
%F 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
%F a(n) = [x^n y^3] Product_{k>=1} 1/(1 - y*x^prime(k)). - _Ilya Gutkovskiy_, Apr 18 2019
%e a(6) = 1 because 6 = 2+2+2,
%e a(9) = 2 because 9 = 2+2+5 = 3+3+3,
%e a(15) = 3 because 15 = 2+2+11 = 3+5+7 = 5+5+5,
%e a(17) = 4 because 17 = 2+2+13 = 3+3+11 = 3+7+7 = 5+5+7.
%e - _Zak Seidov_, Jun 29 2017
%t 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 *)
%t 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 *)
%o (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
%o (Python)
%o from sympy import isprime, primerange, floor
%o def a(n):
%o s=0
%o for p in primerange(((n + 2)//3), n - 3):
%o for q in primerange(((n - p + 1)//2), min(n - p - 2, p) + 1):
%o if isprime(n - p - q): s+=1
%o return s
%o print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jul 01 2017, after PARI code by _Charles R Greathouse IV_
%Y Bisections: A045917, A054860. Cf. A002375, A007963, A061358, A059998.
%Y First occurrence: A139321. Records: A139322.
%Y Column k=3 of A117278.
%K easy,nonn
%O 1,9
%A _Naohiro Nomoto_, Feb 24 2002
%E More terms from _Vladeta Jovovic_, Mar 10 2002
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