A235645 - OEIS

%I #8 Jan 13 2014 16:56:35

%S 0,1,1,1,2,1,2,2,2,2,3,3,3,2,3,3,5,4,3,3,4,5,5,5,5,5,5,4,6,6,5,6,6,4,

%T 6,7,8,8,8,7,9,8,8,7,9,8,9,8,6,9,9,10,10,9,9,12,12,11,13,12,10,11,11,

%U 9,12,11,12,11,11,12,15,14,12,12,12

%N From Goldbach's conjecture and Chen's theorem: number of decompositions of 2n as the sum of either two primes, or a prime and a semiprime.

%C The first 15 terms from this sequence and from A045917 are identical.

%H Jean-François Alcover, <a href="/A235645/b235645.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/ChensTheorem.html">Chen's theorem</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Chen&#39;s_theorem">Chen's theorem</a>

%e 40 = 23+17 = 29+11 = 37+3, so a(20) = 3.

%e Compare with 40 = 23+17 = 29+11 = 31+9 = 37+3 and A045917(20) = 4.

%t a[n_] := Count[IntegerPartitions[2*n, {2}], {p_, q_} /; PrimeQ[p] && (PrimeQ[q] || Length[FactorInteger[q]] == 2)]; Table[a[n], {n, 1, 100}]

%Y Cf. A002375, A045917.

%K nonn

%O 1,5

%A _Jean-François Alcover_, Jan 13 2014