A269329 Number of partitions of a positive integer n into two distinct primes such that for even n, it is of the form n = p + q and for odd n, it is of the form n = 2p' + q'.

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 4, 2, 2, 2, 2, 2, 3, 3, 3, 2, 4, 2, 4, 3, 2, 2, 4, 3, 3, 4, 4, 1, 3, 3, 5, 4, 4, 3, 6, 3, 4, 5, 4, 4, 6, 3, 5, 5, 3, 3, 6, 3, 3, 6, 3, 2, 7, 5, 7, 6, 5, 2, 6, 5, 4, 6, 4, 4, 8, 5, 7, 7, 5, 4, 8, 4, 4, 8, 7, 4, 7, 4, 7, 9, 4, 4, 10, 4, 5, 7, 6, 3, 9

Offset: 1

Author

Frank M Jackson and M. B. Rees, Feb 23 2016

Keywords

• nonn

Comments

This sequence combines Levy's conjecture for odd positive numbers with the Goldbach conjecture for even positive numbers and strenghtens both by restricting the prime pairs to be distinct. I.e., every positive integer n > 6 is the sum of two distinct primes p and q such that for n even, it is of the form n = p + q and for n odd, it is of the form n = 2p' + q'.

Examples

			a(23)=3. Hence there are 3 partitions (as defined above) of the odd integer 23, namely 19+2+2, 17+3+3 and 13+5+5. a(24)=3. Hence there are 3 partitions of the even integer 24, namely 19+5, 17+7 and 13+11.
		

Crossrefs

Cf. A002375, A061357, A194830, A223893, A225522.

Programs

Mathematica

parts[n_, a_, b_] := Select[IntegerPartitions[n, {a+b}, Prime@Range[PrimePi[n]]], Length[Union@#]==2&&MemberQ[Values@Counts@#, a] &]; lst1=Table[Length@parts[2n-1, 1, 2], {n, 1, 200}]; lst2=Table[Length@parts[2n, 1, 1], {n, 1, 200}]; Riffle[lst1, lst2]