A224708 The number of unordered partitions {a,b} of n such that a and b are composite.

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

Offset

1,12

Comments

For n > 11, a(n) > 0. - Geoffrey Critzer, Jan 31 2015

Last occurrence of n is a(A014092(n+4)). - Anthony Browne, May 25 2016

Links

J. Stauduhar, Table of n, a(n) for n = 1..10000

Formula

a(2*n) - a(2*n+1) + A010051(n) = A045917(n). - Anthony Browne, May 03 2016

a(A014092(n+4)) = n. - Anthony Browne, May 25 2016

Example

For n=8, in the set {{7,1},{6,2},{5,3},{4,4}}, {4,4} is the only partition {a,b} where a and b are both composite, so a(8)=1.
For n=12, we have partitions {8,4} and {6,6}, so a(12)=2.

Mathematica

nn = 76; Rest[Transpose[CoefficientList[Series[Product[1/(1 - y x^i), {i, Select[Range[2, nn], ! PrimeQ[#] &]}], {x, 0, nn}], {x, y}]][[3]]] (* Geoffrey Critzer, Jan 31 2015 *)
f[n_] := Count[ PrimeQ@ Rest@ IntegerPartitions[ n, {2}], {False, False}]; Array[f, 76] (* Robert G. Wilson v, Feb 04 2015 *)

Crossrefs

Cf. A010051, A014092, A045917, A066247.

Sequence in context: A326850 A303756 A105259 * A322023 A029229 A029216

Adjacent sequences: A224705 A224706 A224707 * A224709 A224710 A224711

Keyword

nonn,easy

Author

J. Stauduhar, Apr 16 2013

Status

approved