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

Author

Wesley Ivan Hurt, Dec 22 2017

Keywords

• nonn
• easy

Examples

			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)
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A023645.

Programs

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 *)

PARI

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

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