A296966 Sum of all the parts in the partitions of n into two distinct parts such that the smaller part divides the larger.

0, 0, 3, 4, 5, 12, 7, 16, 18, 20, 11, 48, 13, 28, 45, 48, 17, 72, 19, 80, 63, 44, 23, 144, 50, 52, 81, 112, 29, 180, 31, 128, 99, 68, 105, 252, 37, 76, 117, 240, 41, 252, 43, 176, 225, 92, 47, 384, 98, 200, 153, 208, 53, 324, 165, 336, 171, 116, 59, 600, 61

Offset

1,3

Links

Table of n, a(n) for n=1..61.

Index entries for sequences related to partitions

Formula

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

a(n) = n * A023645(n). - Robert G. Wilson v, Dec 24 2017

Example

From Wesley Ivan Hurt, Feb 21 2018: (Start)
a(5) = 5; there is one partition of 5 into two distinct parts such that the smaller part divides the larger, namely (4,1), so the sum of the parts is then 4 + 1 = 5.
a(6) = 12; the partitions of 6 into two distinct parts such that the smaller part divides the larger are (5,1) and (4,2), and the sum of the parts is then 5 + 1 + 4 + 2 = 12.
a(7) = 7; there is one partition of 7 into two distinct parts such that the smaller part divides the larger, namely (6,1), so the sum of the parts is 6 + 1 = 7.
a(8) = 16; there are two partitions of 8 into 2 distinct parts such that the smaller divides the larger, namely (7,1) and (6,2). The sum of the parts is then 7 + 1 + 6 + 2 = 16.
(End)

Mathematica

Table[n*Sum[(Floor[n/i] - Floor[(n - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 100}]
f[n_] := n*Length[Select[Divisors@n, 2 # < n &]]; Array[f, 61] (* or *)
f[n_] := Block[{t = DivisorSigma[0, n]}, n*If[OddQ@ n, t -1, t -2]]; Array[f, 61] (* Robert G. Wilson v, Dec 24 2017 *)

Prog

(PARI) a(n) = n*sum(i=1, floor((n-1)/2), floor(n/i) - floor((n-1)/i)) \\ Iain Fox, Dec 22 2017

Crossrefs

Cf. A023645.

Sequence in context: A173096 A302752 A046964 * A235598 A191197 A055493

Adjacent sequences: A296963 A296964 A296965 * A296967 A296968 A296969

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 22 2017

Status

approved