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

1,9

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to partitions.

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).

Example

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).

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}]

Prog

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

Crossrefs

Cf. A000720, A010051, A051699.

Sequence in context: A175599 A179759 A257260 * A159195 A265859 A271420

Adjacent sequences: A334205 A334206 A334207 * A334209 A334210 A334211

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 18 2020

Extensions

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

Status

approved