A224708 - OEIS

%I #37 Feb 01 2021 18:17:38

%S 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,

%T 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,

%U 13,15,13,18,13,20,14,17,15,20,15,22,16,20,16,21

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

%C For n > 11, a(n) > 0. - _Geoffrey Critzer_, Jan 31 2015

%C Last occurrence of n is a(A014092(n+4)). - _Anthony Browne_, May 25 2016

%H J. Stauduhar, <a href="/A224708/b224708.txt">Table of n, a(n) for n = 1..10000</a>

%F a(2*n) - a(2*n+1) + A010051(n) = A045917(n). - _Anthony Browne_, May 03 2016

%F a(A014092(n+4)) = n. - _Anthony Browne_, May 25 2016

%e 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.

%e For n=12, we have partitions {8,4} and {6,6}, so a(12)=2.

%t 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 *)

%t f[n_] := Count[ PrimeQ@ Rest@ IntegerPartitions[ n, {2}], {False, False}]; Array[f, 76] (* _Robert G. Wilson v_, Feb 04 2015 *)

%Y Cf. A010051, A014092, A045917, A066247.

%K nonn,easy

%O 1,12

%A _J. Stauduhar_, Apr 16 2013