A294013 Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the smaller part prime.

0, 0, 2, 6, 10, 16, 22, 30, 38, 46, 54, 64, 74, 86, 98, 110, 122, 136, 150, 166, 182, 198, 214, 232, 250, 268, 286, 304, 322, 342, 362, 384, 406, 428, 450, 472, 494, 518, 542, 566, 590, 616, 642, 670, 698, 726, 754, 784, 814, 844, 874, 904, 934, 966, 998

Offset

1,3

Comments

Sum of the slopes of the tangent lines along the left side of the parabola b(x) = 2*n*x-x^2 at prime values of x for x in 0 < x <= n. For example, d/dx 2*n*x-x^2 = 2n-2x. So for a(6), the prime values of x are x=2,3,5 and so a(6) = 12-2*2 + 12-2*3 + 12-2*5 = 8 + 6 + 2 = 16. - Wesley Ivan Hurt, Mar 24 2018

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Index entries for sequences related to partitions

Formula

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

Example

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

Mathematica

Table[2 Sum[(n - i) (PrimePi[i] - PrimePi[i - 1]), {i, n}], {n, 40}]
Table[Total[#[[1]]-#[[2]]&/@Select[IntegerPartitions[2n, {2}], PrimeQ[ #[[2]]]&]], {n, 60}] (* Harvey P. Dale, Jan 25 2019 *)

Crossrefs

Cf. A010051, A290288.

Sequence in context: A220453 A195957 A354425 * A183575 A096184 A254829

Adjacent sequences: A294010 A294011 A294012 * A294014 A294015 A294016

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 21 2017

Status

approved