A224709 - OEIS

%I #16 Apr 11 2022 20:47:47

%S 0,0,0,1,1,2,2,3,4,4,4,6,5,6,8,7,8,10,8,10,12,11,11,14,13,13,16,14,15,

%T 19,15,18,20,17,20,22,20,21,24,22,22,27,23,24,30,25,26,30,27,30,33,30,

%U 30,34,32,33,37,33,33,41,33,36,42,36,40,43,39,40,44

%N The number of unordered partitions {a,b} of the even numbers 2n such that a and b are composite.

%C Conjecture: a(3n+9) > a(3n+8) and a(3n+10) < a(3n+9) for n>=1. - _Anthony Browne_, Jun 26 2016

%H J. Stauduhar, <a href="/A224709/b224709.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{k=1..n-1} (1-A010051(k+1))(1-A010051(2n-k-1)). - _Anthony Browne_, Jun 26 2016

%e For n=6, 2*6=12 and the partitions of 12 are (1,11),(2,10),(3,9),(4,8),(5,7),(6,6). Of these, 2 are composite pairs, namely (4,8),(6,6) so a(6)=2.

%t Table[Count[Transpose@ {#, 2 n - #} &@ Range@ n, w_ /; Times @@ Boole@ Map[CompositeQ, w] > 0], {n, 69}] (* _Michael De Vlieger_, Jun 26 2016 *)

%o (PARI) a(n) = sum(k=1, n-1, (1-isprime(k+1))*(1-isprime(2*n-k-1))); \\ _Michel Marcus_, Apr 11 2022

%Y Subsequence of A224708.

%Y Cf. A010051.

%K nonn

%O 1,6

%A _J. Stauduhar_, Apr 16 2013