A235645 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.

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, 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, 9, 12, 11, 12, 11, 11, 12, 15, 14, 12, 12, 12

Offset: 1

Author

Jean-François Alcover, Jan 13 2014

Keywords

• nonn

Comments

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

Examples

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

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..10000
• Eric Weisstein's MathWorld, Chen's theorem
• Wikipedia, Chen's theorem

Crossrefs

Cf. A002375, A045917.

Programs

Mathematica

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