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A296955 Sum of the smaller parts of the partitions of n into two distinct parts such that the smaller part divides the larger. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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.

Sequence in context: A296388 A081772 A204217 * A219525 A050121 A029152

Adjacent sequences:  A296952 A296953 A296954 * A296956 A296957 A296958

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Dec 22 2017

EXTENSIONS

More terms from Antti Karttunen, Sep 25 2018

STATUS

approved

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Last modified June 25 22:13 EDT 2022. Contains 354868 sequences. (Running on oeis4.)