A339927 Number of partitions of n into two composite parts with the same number of divisors.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 2, 0, 2, 0, 2, 1, 1, 1, 2, 1, 0, 1, 2, 3, 3, 1, 4, 1, 2, 2, 4, 2, 1, 1, 4, 4, 3, 3, 4, 2, 2, 3, 7, 4, 3, 0, 4, 4, 5, 2, 5, 3, 1, 3, 7, 6, 3, 3, 6, 6, 5, 3, 6, 2, 6, 3, 11, 7, 2, 3, 4, 6, 5, 5, 8, 3, 4, 5, 10, 4, 4, 3, 7, 5, 7, 7, 7

Offset: 1

Author

Wesley Ivan Hurt, Dec 23 2020

Keywords

• nonn

Examples

			a(18) = 2; 18 has two partitions into two composite parts that have the same number of divisors, (10,8) and (9,9).
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A000005, A066247, A224708.

Programs

Mathematica

Table[Sum[KroneckerDelta[DivisorSigma[0, i], DivisorSigma[0, n - i]] (1 - PrimePi[i] + PrimePi[i - 1]) (1 - PrimePi[n - i] + PrimePi[n - i - 1]), {i, 2, Floor[n/2]}], {n, 100}]
Table[Count[IntegerPartitions[n,{2}],?(AllTrue[#,CompositeQ]&&Length[Union[ DivisorSigma[ 0,#]]]==1&)],{n,100}] (* _Harvey P. Dale, Jul 02 2022 *)

Formula

a(n) = Sum_{k=2..floor(n/2)} [d(k) = d(n-k)] * c(k) * c(n-k), where [ ] is the Iverson bracket, d(n) is the number of divisors of n (A000005), and c is the characteristic function of composite numbers (A066247).