A119707 Number of distinct primes appearing in all partitions of n into prime parts.

0, 1, 1, 1, 3, 2, 4, 3, 4, 4, 5, 4, 6, 5, 6, 6, 7, 6, 8, 7, 8, 8, 9, 8, 9, 9, 9, 9, 10, 9, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 13, 12, 14, 13, 14, 14, 15, 14, 15, 15, 15, 15, 16, 15, 16, 16, 16, 16, 17, 16, 18, 17, 18, 18, 18, 18, 19, 18, 19, 19, 20, 19, 21, 20, 21, 21, 21, 21, 22

Offset: 1

Author

Anton Joha, Jun 10 2006

Keywords

• nonn

Examples

			There is only 1 distinct prime number involved in the partitions of 4, namely 2 (in 2+2 = 4). The partition 3+1 does not count, as 1 is not a prime. So a(4)= 1.
There are 3 distinct primes involved in the partitions of 5 = 2+3, so a(5) = 3.
		

Crossrefs

Cf. A000720.

Programs

Mathematica

f[n_] := If[OddQ@n, If[n == 3, 1, PrimePi@n], If[n == 2, 1, PrimePi[n - 2]]]; Array[f, 80] (* Robert G. Wilson v *)

Formula

When n = odd and >=5 then a(n) = pi(n) = A000720(n). When n = even and >=4 then a(n) = pi(n-2) = A000720(n-2)

Extensions

Edited and extended by Robert G. Wilson v, Jun 15 2006