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

1,6

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

Links

Table of n, a(n) for n=1..59.

Index entries for sequences related to partitions

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

Example

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

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}]

Prog

(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

Crossrefs

Cf. A010051, A056172, A294022, A000720.

Sequence in context: A165453 A282168 A025224 * A096182 A186347 A302648

Adjacent sequences: A294020 A294021 A294022 * A294024 A294025 A294026

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 21 2017

Status

approved