A297024 -id:A297024 - cp's OEIS frontend

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

Offset: 1

Wesley Ivan Hurt, Dec 22 2017

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

			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.

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

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

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

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 *)

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

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

More terms from Antti Karttunen, Sep 25 2018

A303384 Total area of all rectangles with dimensions s and t where s | t, n = s + t and s <= t.

Original entry on oeis.org

0, 1, 2, 7, 4, 22, 6, 35, 26, 50, 10, 126, 12, 86, 100, 155, 16, 247, 18, 294, 172, 182, 22, 590, 124, 242, 260, 518, 28, 860, 30, 651, 364, 386, 380, 1365, 36, 470, 484, 1390, 40, 1532, 42, 1134, 1144, 662, 46, 2542, 342, 1395, 772, 1526, 52, 2380, 788

Offset: 1

Wesley Ivan Hurt, Apr 22 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions

Cf. A297024, A303385.
Cf. A000203, A001157.
Cf. A002117, A013661.

List([1..60],n->Sum([1..Int(n/2)],i->i*(n-i)*(Int((n-i)/i)-Int((n-i-1)/i)))); # Muniru A Asiru, Jun 07 2018

[0] cat [&+[k*(n-k)*((n-k) div k)-(n-k-1) div k:  k in [1..n div 2]]: n in [2..80]]; // Vincenzo Librandi, Jun 07 2018

with(numtheory): seq(n*sigma(n) - sigma[2](n), n=1..60); # Ridouane Oudra, Apr 15 2021

Table[Sum[i (n - i) (Floor[(n - i)/i] - Floor[(n - i - 1)/i]), {i, Floor[n/2]}], {n, 80}]
a[n_] := n * DivisorSigma[1, n] - DivisorSigma[2, n]; Array[a, 100] (* Amiram Eldar, Dec 11 2023 *)

a(n) = sum(i=1, n\2, i*(n-i)*((n-i)\i - (n-i-1)\i)); \\ Michel Marcus, Jun 07 2018

a(n) = sumdiv(n, d, d*(n-d)); \\ Daniel Suteu, Jun 19 2018

a(n) = {my(f = factor(n)); n * sigma(f) - sigma(f, 2);} \\ Amiram Eldar, Dec 11 2023

a(n) = Sum_{i=1..floor(n/2)} i * (n-i) * (floor((n-i)/i) - floor((n-i-1)/i)).
a(n) = Sum_{d|n} d*(n-d). - Daniel Suteu, Jun 19 2018
a(n) = n*sigma(n) - sigma_2(n). - Ridouane Oudra, Apr 15 2021
From Amiram Eldar, Dec 11 2023: (Start)
a(n) = A064987(n) - A001157(n).
Sum_{k=1..n} a(k) ~  c * n^3 / 3, where c = zeta(2) - zeta(3) = 0.442877... . (End)

A297053 Sum of the larger parts of the partitions of n into two parts such that the smaller part does not divide the larger.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 9, 5, 12, 13, 30, 7, 45, 38, 41, 43, 84, 48, 108, 67, 103, 124, 165, 78, 178, 185, 192, 175, 273, 162, 315, 247, 308, 343, 350, 244, 459, 440, 451, 360, 570, 411, 630, 535, 545, 670, 759, 496, 786, 718, 818, 787, 975, 768, 959, 834, 1042

Offset: 1

Wesley Ivan Hurt, Dec 24 2017

			a(10) = 13; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4) and (5,5). The sum of the larger parts of these partitions such that the smaller part does not divide the larger is then 7 + 6 = 13.

Index entries for sequences related to partitions

Cf. A297024.

Table[Sum[(n - i) (1 - (Floor[n/i] - Floor[(n - 1)/i])), {i, Floor[n/2]}], {n, 80}]

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

cp's OEIS Frontend

A296955 Sum of the smaller parts of the partitions of n into two distinct parts such that the smaller part divides the larger.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A303384 Total area of all rectangles with dimensions s and t where s | t, n = s + t and s <= t.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

PARI

Formula

A297053 Sum of the larger parts of the partitions of n into two parts such that the smaller part does not divide the larger.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula