A296955 - OEIS

A296955

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

0, 0, 1, 1, 1, 3, 1, 3, 4, 3, 1, 10, 1, 3, 9, 7, 1, 12, 1, 12, 11, 3, 1, 24, 6, 3, 13, 14, 1, 27, 1, 15, 15, 3, 13, 37, 1, 3, 17, 30, 1, 33, 1, 18, 33, 3, 1, 52, 8, 18, 21, 20, 1, 39, 17, 36, 23, 3, 1, 78, 1, 3, 41, 31, 19, 45, 1, 24, 27, 39, 1, 87, 1, 3, 49, 26, 19, 51, 1, 66, 40, 3, 1, 98, 23, 3, 33, 48, 1, 99, 21, 30, 35, 3, 25, 108, 1

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OFFSET

1,6

COMMENTS

The number of partitions of n into 3 parts whose "middle" part divides n. - Wesley Ivan Hurt, Oct 21 2021

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Index entries for sequences related to partitions

FORMULA

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

a(n) = the sum of the divisors < n/2. - Robert G. Wilson v, Dec 23 2017

a(n) = 1 iff n is an odd prime or n=4. - Robert G. Wilson v, Dec 23 2017

G.f.: Sum_{k>=1} k * x^(3*k) / (1 - x^k). - Ilya Gutkovskiy, May 30 2020

G.f.: Sum_{k >= 3} x^k/(1 - x^k)^2. Cf. A023645. - Peter Bala, Jan 13 2021

Faster converging g.f.: Sum_{n >= 1} q^(n*(n+2))*( n*q^(3*n+4) - (n + 1)*q^(2*n+2) - (n - 1)*q^(n+2) + n )/( (1 - q^n )*(1 - q^(n+2))^2 ). (In equation 1 in Arndt, after combining the two n = 0 summands to get t/(1 - t), apply the operator t*d/dt and then set t = q^2 and x = 1. Cf. A001065.) - Peter Bala, Jan 22 2021

a(n) = A000203(n) - A080512(n). - Ridouane Oudra, Aug 15 2024

EXAMPLE

a(12) = 10; the partitions of 12 into two distinct parts are (11,1), (10,2), (9,3), (8,4) and (7,5). 1 divides 11, 2 divides 10, 3 divides 9 and 4 divides 8, so the sum of the smaller parts gives 1 + 2 + 3 + 4 = 10.

MAPLE

with(numtheory):

a := n -> add( d, d = divisors(n) minus {floor((n+1)/2), n} ):

seq(a(n), n = 1..100); # Peter Bala, Jan 13 2021

MATHEMATICA

Table[Sum[i (Floor[n/i] - Floor[(n - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 100}]

f[n_] := Plus @@ Select[Divisors@n, 2 # < n &]; Array[f, 75] (* Robert G. Wilson v, Dec 23 2017 *)

PROG

(PARI) A296955(n) = sumdiv(n, d, (d<(n/2))*d); \\ Antti Karttunen, Sep 25 2018

CROSSREFS

Cf. A297024, A001065, A023645, A000203, A080512.

Sequence in context: A296388 A081772 A204217 * A219525 A050121 A029152