A290288 Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the larger part prime.

0, 2, 4, 8, 4, 12, 20, 16, 28, 40, 32, 48, 40, 34, 56, 78, 68, 60, 88, 80, 112, 144, 132, 168, 156, 144, 184, 170, 156, 202, 248, 234, 220, 272, 256, 310, 364, 346, 328, 388, 368, 432, 412, 394, 464, 444, 424, 406, 484, 464, 544, 624, 600, 684, 768, 742, 828

Offset

1,2

Comments

Sum of the slopes of the tangent lines along the left side of the parabola b(x) = 2*n*x-x^2 such that 2n-x is prime for integer values of x in 0 < x <= n. For example, d/dx 2*n*x-x^2 = 2n-2x. So for a(8), the integer values of x which make 16-x prime are x=3,5 and so a(8) = 16-2*3 + 16-2*5 = 10 + 6 = 16. - Wesley Ivan Hurt, Mar 24 2018

Links

Table of n, a(n) for n=1..57.

Index entries for sequences related to partitions

Formula

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

Example

a(4) = 8; there are two partitions of 2*4 = 8 into two parts with the larger part prime: (7,1) and (5,3). The sum of the differences of the parts is (7 - 1) + (5 - 3) = 6 + 2 = 8.

Mathematica

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

Prog

(PARI) a(n) = 2*sum(i=1, n, (n-i)*isprime(2*n-i)); \\ Michel Marcus, Mar 25 2018

Crossrefs

Cf. A010051, A294013.

Sequence in context: A016635 A133992 A294094 * A126215 A165617 A273170

Adjacent sequences: A290285 A290286 A290287 * A290289 A290290 A290291

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 21 2017

Status

approved