A334208 Number of partitions of 2n into two composite parts, (r,s), such that r and s have the same number of primes less than or equal to them.

0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2

Offset: 1

Author

Wesley Ivan Hurt, Apr 18 2020

Keywords

• nonn
• easy

Comments

Apparently a(n) = A051699(n) for n>=2. - R. J. Mathar, Apr 22 2020

Examples

			a(9) = 2; 2*9 = 18 has two partitions into composite parts, (10,8) and (9,9), such that pi(10) = 4 = pi(8) and pi(9) = 4 = pi(9).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to partitions.

Crossrefs

Cf. A000720, A010051, A051699.

Programs

Mathematica

Table[Sum[KroneckerDelta[PrimePi[i], PrimePi[2 n - i]] (1 - PrimePi[i] + PrimePi[i - 1]) (1 - PrimePi[2 n - i] + PrimePi[2 n - i - 1]), {i, 2, n}], {n, 100}]

PARI

A334208(n) = sum(i=2,n,(!isprime(i) && !isprime(n+n-i) && primepi(i)==primepi(n+n-i))); \\ Antti Karttunen, Jan 29 2025

Formula

a(n) = Sum_{i=2..n} [pi(i) = pi(2*n-i)] * (1 - c(i)) * (1 - c(2*n-i)), where [] is the Iverson bracket, pi is the prime counting function (A000720), and c is the prime characteristic (A010051).

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 29 2025