A294023 Sum of the differences of the larger and smaller parts in the partitions of n into two parts with the smaller part prime.

0, 0, 0, 0, 1, 2, 4, 6, 8, 10, 13, 16, 19, 22, 26, 30, 34, 38, 42, 46, 50, 54, 59, 64, 69, 74, 80, 86, 92, 98, 104, 110, 116, 122, 129, 136, 143, 150, 158, 166, 174, 182, 190, 198, 206, 214, 223, 232, 241, 250, 259, 268, 277, 286, 295, 304, 313, 322, 332

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 at prime values of x for x in 0 < x <= floor(n/2). For example, d/dx n*x-x^2 = n-2x. So for a(11), x=2,3,5 and so 11-2*2 + 11-2*3 + 11-2*5 = 7 + 5 + 1 = 13. - Wesley Ivan Hurt, Mar 24 2018

Examples

			The partitions of n = 11 into a number and a smaller prime number are 9 + 2, 8 + 3, and 6 + 5, so a(11) = (9 - 2) + (8 - 3) + (6 - 5) = 13. - _Michael B. Porter_, Apr 06 2018
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A056172, A294022, A000720.

Programs

Maple

with(numtheory): seq(add(pi(floor(i/2)), i=1..n-1), n=1..100); # Ridouane Oudra, Nov 24 2019

Mathematica

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

PARI

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

PARI

a(n) = my(res = 0); forprime(p = 2, n >> 1, res += (n - p << 1)); res \\ David A. Corneth, Apr 06 2018

Formula

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

First differences are A056172. - David A. Corneth, Apr 06 2018

a(n) = Sum_{i=1..n-1} pi(floor(i/2)), where pi(n) = A000720(n). - Ridouane Oudra, Nov 24 2019