A345039 - OEIS

%I #4 Jun 06 2021 13:55:44

%S 0,0,0,0,0,0,0,1,0,1,0,2,0,2,1,3,0,4,0,4,2,4,0,6,1,5,3,6,0,8,0,7,4,7,

%T 3,10,0,8,5,10,0,12,0,10,8,10,0,14,2,13,7,12,0,16,5,14,8,13,0,19,0,14,

%U 11,15,6,20,0,16,10,20,0,22,0,17,15,18,6,24,0,22,12,19,0,27

%N Number of partitions of n into two composite parts that share a nontrivial divisor.

%F a(n) = Sum_{k=1..floor(n/2)} (1 - floor(1/gcd(k,n-k))) * c(k) * c(n-k), where c(n) is the characteristic function of composite numbers.

%e a(12) = 2; (8,4) and (6,6).

%e a(15) = 1; (9,6).

%e a(16) = 3; (12,4), (10,6), and (8,8).

%e a(18) = 4; (14,4), (12,6), (10,8), and (9,9).

%t Table[Sum[(1 - PrimePi[k] + PrimePi[k - 1]) (1 - PrimePi[n - k] + PrimePi[n - k - 1]) (1 - Floor[1/GCD[k, n - k]]), {k, Floor[n/2]}], {n, 100}]

%Y Cf. A062610, A066247.

%K nonn

%O 1,12

%A _Wesley Ivan Hurt_, Jun 06 2021