A294022 Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the larger part prime.

0, 0, 1, 2, 1, 4, 3, 8, 6, 4, 3, 12, 10, 20, 18, 16, 14, 28, 25, 40, 36, 32, 29, 48, 44, 40, 37, 34, 31, 56, 52, 78, 73, 68, 64, 60, 56, 88, 84, 80, 76, 112, 107, 144, 138, 132, 127, 168, 162, 156, 150, 144, 138, 184, 177, 170, 163, 156, 150, 202, 195, 248

Offset: 1

Author

Wesley Ivan Hurt, Oct 21 2017

Keywords

• nonn
• easy

Comments

Sum of the slopes of the tangent lines along the left side of the parabola b(x) = n*x-x^2 such that n-x is prime for x in 0 < x <= floor(n/2). For example, d/dx n*x-x^2 = n-2x. So for a(12), the integer values of x which make 12-x prime are x=1,5 and so a(12) = 12-2*1 + 12-2*5 = 10 + 2 = 12. - Wesley Ivan Hurt, Mar 24 2018

Examples

			There are two ways to partition n = 9 into a prime and a smaller positive integer: 7 + 2 and 5 + 4. So a(9) = (7 - 2) + (5 - 4) = 6. - _Michael B. Porter_, Mar 26 2018
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A294023.

Programs

Mathematica

Table[Sum[(n - 2 i) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[n/2]}], {n, 60}]

PARI

a(n) = sum(i=1, n\2, (n - 2*i)*isprime(n-i)); \\ Michel Marcus, Mar 24 2018

Formula

a(n) = Sum_{i=1..floor(n/2)} (n - 2i) * A010051(n - i).