

A296966


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


1



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
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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((n1)/2)} floor(n/i)  floor((n1)/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((n1)/2), floor(n/i)  floor((n1)/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



