A297024 Sum of the smaller parts of the partitions of n into two parts such that the smaller part does not divide the larger.

0, 0, 0, 0, 2, 0, 5, 3, 6, 7, 14, 5, 20, 18, 19, 21, 35, 24, 44, 33, 44, 52, 65, 42, 72, 75, 78, 77, 104, 78, 119, 105, 121, 133, 140, 116, 170, 168, 173, 160, 209, 177, 230, 213, 220, 250, 275, 224, 292, 282, 304, 305, 350, 312, 361, 342, 383, 403, 434, 357

Offset

1,5

Links

Muniru A Asiru, Table of n, a(n) for n = 1..1000

Index entries for sequences related to partitions

Formula

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

Example

a(10) = 7; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4), (5,5). The sum of the smaller parts that do not divide their larger counterparts is then 3 + 4 = 7.

Mathematica

Table[Sum[i (1 - (Floor[n/i] - Floor[(n - 1)/i])), {i, Floor[n/2]}], {n, 100}]
f[n_] := Plus @@ Select[ Range[n/2], !MemberQ[Divisors[n], #] &]; Array[f, 60] (* Robert G. Wilson v, Dec 23 2017 *)
Table[Total[Select[IntegerPartitions[n, {2}], Mod[#[[1]], #[[2]]]!=0&][[All, 2]]], {n, 60}] (* Harvey P. Dale, Dec 17 2021 *)

Prog

(GAP) List(List(List([1..10^2], n-> Partitions(n, 2)), i -> Filtered(i, j -> j[1] mod j[2] <> 0)), m->Sum(m, k -> k[2])); # Muniru A Asiru, Jan 28 2018

Crossrefs

Cf. A296955.

Sequence in context: A121705 A071782 A328495 * A356215 A107363 A154954

Adjacent sequences: A297021 A297022 A297023 * A297025 A297026 A297027

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 23 2017

Status

approved