A362831 Number of partitions of n into two distinct parts (s,t) such that pi(s) = pi(t).

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 2, 2, 3, 2, 2, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 2, 3, 2, 2, 1, 1, 0, 0, 0, 1, 1, 2, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 1

Offset: 1

Author

Wesley Ivan Hurt, May 04 2023

Keywords

• nonn
• easy

Examples

			a(51) = 3. The 3 partitions of 51 are (23,28), (24,27), and (25,26).
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A000720 (pi), A362721.

Programs

Mathematica

Table[Sum[KroneckerDelta[PrimePi[k], PrimePi[n - k]], {k, Floor[(n - 1)/2]}], {n, 100}]

Formula

a(n) = Sum_{k=1..floor((n-1)/2)} [pi(k) = pi(n-k)], where [ ] is the Iverson bracket and pi is the prime counting function (A000720).

a(n) = (A362721(n-1) - ((n-1) mod 2))/2.