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

1,16

Links

Table of n, a(n) for n=1..92.

Index entries for sequences related to partitions

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).

Example

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

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 *)

Crossrefs

Cf. A000005, A066247, A224708.

Sequence in context: A230632 A052511 A054533 * A227957 A305575 A247977

Adjacent sequences: A339924 A339925 A339926 * A339928 A339929 A339930

Keyword

nonn

Author

Wesley Ivan Hurt, Dec 23 2020

Status

approved