A152211 -id:A152211 - cp's OEIS frontend

A094471 a(n) = Sum_{(n - k)|n, 0 <= k <= n} k.

0, 1, 2, 5, 4, 12, 6, 17, 14, 22, 10, 44, 12, 32, 36, 49, 16, 69, 18, 78, 52, 52, 22, 132, 44, 62, 68, 112, 28, 168, 30, 129, 84, 82, 92, 233, 36, 92, 100, 230, 40, 240, 42, 180, 192, 112, 46, 356, 90, 207, 132, 214, 52, 312, 148, 328, 148, 142, 58, 552, 60

Offset: 1

Labos Elemer, May 28 2004

Not all values arise and some arise more than once.
Row sums of triangle A134866. - Gary W. Adamson, Nov 14 2007
Sum of the largest parts of the partitions of n into two parts such that the smaller part divides the larger. - Wesley Ivan Hurt, Dec 21 2017
a(n) is also the sum of all parts minus the total number of parts of all partitions of n into equal parts (an interpretation of the Torlach Rush's formula). - Omar E. Pol, Nov 30 2019
If and only if sigma(n) divides a(n), then n is one of Ore's Harmonic numbers, A001599. - Antti Karttunen, Jul 18 2020

			q^2 + 2*q^3 + 5*q^4 + 4*q^5 + 12*q^6 + 6*q^7 + 17*q^8 + 14*q^9 + ...
For n = 4 the partitions of 4 into equal parts are [4], [2,2], [1,1,1,1]. The sum of all parts is 4 + 2 + 2 + 1 + 1 + 1 + 1 = 12. There are 7 parts, so a(4) = 12 - 7 = 5. - _Omar E. Pol_, Nov 30 2019

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 30.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000203, A001599, A038040, A134866, A152211, A244051, A324121 (= gcd(a(n), sigma(n))).

Cf. A088827 (positions of odd terms).

using AbstractAlgebra
function A094471(n)
    sum(k for k in 0:n if is_divisible_by(n, n - k))
end
[A094471(n) for n in 1:61] |> println  # Peter Luschny, Nov 14 2023

with(numtheory); A094471:=n->n*tau(n)-sigma(n); seq(A094471(k), k=1..100); # Wesley Ivan Hurt, Oct 27 2013
divides := (k, n) -> k = n or (k > 0 and irem(n, k) = 0):
a := n -> local k; add(`if`(divides(n - k, n), k, 0), k = 0..n):
seq(a(n), n = 1..61);  # Peter Luschny, Nov 14 2023

Table[n*DivisorSigma[0, n] - DivisorSigma[1, n], {n, 1, 100}]

{a(n) = n*numdiv(n) - sigma(n)} /* Michael Somos, Jan 25 2008 */

from math import prod
from sympy import factorint
def A094471(n):
    f = factorint(n).items()
    return n*prod(e+1 for p,e in f)-prod((p**(e+1)-1)//(p-1) for p,e in f)
# Chai Wah Wu, Nov 14 2023

def A094471(n): return sum(k for k in (0..n) if (n-k).divides(n))
print([A094471(n) for n in range(1, 62)])  # Peter Luschny, Nov 14 2023

a(n) = n*tau(n) - sigma(n) = n*A000005(n) - A000203(n). [Previous name.]
If p is prime, then a(p) = p*tau(p)-sigma(p) = 2p-(p+1) = p-1 = phi(p).
If n>1, then a(n)>0.
a(n) = Sum_{d|n} (n-d). - Amarnath Murthy, Jul 31 2005
G.f.: Sum_{k>=1} k*x^(2*k)/(1 - x^k)^2. - Ilya Gutkovskiy, Oct 24 2018
a(n) = A038040(n) - A000203(n). - Torlach Rush, Feb 02 2019

Simpler name by Peter Luschny, Nov 14 2023

A328203 Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.

Original entry on oeis.org

1, 2, 5, 4, 8, 10, 11, 8, 20, 16, 17, 20, 20, 22, 42, 16, 26, 40, 29, 32, 58, 34, 35, 40, 53, 40, 74, 44, 44, 84, 47, 32, 90, 52, 94, 80, 56, 58, 106, 64, 62, 116, 65, 68, 174, 70, 71, 80, 102, 106, 138, 80, 80, 148, 146, 88, 154, 88, 89, 168, 92, 94, 241, 64, 172

Offset: 1

Ilya Gutkovskiy, Oct 07 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A328203, Sequence Machine.

Cf. A000005, A000079 (fixed points), A000203, A001227, A002131, A003602, A006519, A006918, A026741, A038040, A109168, A113415, A140472, A152211, A193356, A245579, A349353 (Dirichlet inverse), A349354.

Cf. also A347957.

a:=[]; for k in [1..65] do if IsOdd(k) then a[k]:=(k * #Divisors(k) + DivisorSigma(1,k)) / 2; else a[k]:=(k * (#Divisors(k) - #Divisors(k div 2)) + DivisorSigma(1,k) - DivisorSigma(1,k div 2)) / 2;  end if; end for; a; // Marius A. Burtea, Oct 07 2019

nmax = 65; CoefficientList[Series[Sum[k x^k/(1 - x^(2 k))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, (n Mod[#, 2] + Boole[OddQ[n/#]] #)/2 &]; Table[a[n], {n, 1, 65}]

A328203(n) = if(n%2,(1/2)*(sigma(n)+(n*numdiv(n))),2*A328203(n/2)); \\ Antti Karttunen, Nov 13 2021

a(n) = (n * d(n) + sigma(n)) / 2 if n odd, (n * (d(n) - d(n/2)) + sigma(n) - sigma(n/2)) / 2 if n even.
a(n) = (n * A001227(n) + A002131(n)) / 2.
a(2*n) = 2 * a(n).
From Antti Karttunen, Nov 13 2021: (Start)
The following two convolutions were found by Jon Maiga's Sequence Machine search algorithm. Both are easy to prove:
a(n) = Sum_{d|n} A003602(d) * A026741(n/d).
a(n) = Sum_{d|n} A109168(d) * A193356(n/d), where A109168(d) = A140472(d) = (d+A006519(d))/2.
(End)

A309732 Expansion of Sum_{k>=1} k^2 * x^k/(1 - x^k)^3.

Original entry on oeis.org

1, 7, 15, 38, 40, 108, 77, 188, 180, 290, 187, 600, 260, 560, 630, 888, 442, 1323, 551, 1620, 1218, 1364, 805, 3024, 1325, 1898, 1998, 3136, 1276, 4680, 1457, 4080, 2970, 3230, 3290, 7470, 2072, 4028, 4134, 8200, 2542, 9072, 2795, 7656, 7830, 5888, 3337, 14496, 4998, 9825, 7038

Offset: 1

Ilya Gutkovskiy, Aug 14 2019

nonn

Dirichlet convolution of triangular numbers (A000217) with squares (A000290).

a(n) is n times half m, where m is the sum of all parts plus the total number of parts of the partitions of n into equal parts. - Omar E. Pol, Nov 30 2019

Marius A. Burtea, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A000217, A000290, A007437, A034714, A034715, A038040, A064987, A152211, A309731, A244051.

[n*(n*NumberOfDivisors(n) + DivisorSigma(1,n))/2:n in [1..51]]; // Marius A. Burtea, Nov 29 2019

with(numtheory): seq(n*(n*tau(n)+sigma(n))/2, n=1..50); # Ridouane Oudra, Nov 28 2019

nmax = 51; CoefficientList[Series[Sum[k^2 x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DirichletConvolve[j (j + 1)/2, j^2, j, n], {n, 1, 51}]
Table[n (n DivisorSigma[0, n] + DivisorSigma[1, n])/2, {n, 1, 51}]

a(n)=sumdiv(n, d, binomial(n/d+1,2)*d^2); \\ Andrew Howroyd, Aug 14 2019

a(n)=n*(n*numdiv(n) + sigma(n))/2; \\ Andrew Howroyd, Aug 14 2019

G.f.: Sum_{k>=1} (k*(k + 1)/2) * x^k * (1 + x^k)/(1 - x^k)^3.
a(n) = n * (n * d(n) + sigma(n))/2.
Dirichlet g.f.: zeta(s-2) * (zeta(s-2) + zeta(s-1))/2.
a(n) = n*(A038040(n) + A000203(n))/2 = n*A152211(n)/2. - Omar E. Pol, Aug 17 2019
a(n) = Sum_{k=1..n} k*sigma(gcd(n,k)). - Ridouane Oudra, Nov 28 2019

A308668 a(n) = Sum_{d|n} d^(n/d+n).

Original entry on oeis.org

1, 9, 82, 1089, 15626, 287010, 5764802, 135270401, 3487315843, 100244173394, 3138428376722, 107072686593858, 3937376385699290, 155601328490478978, 6568412173896940652, 295165920677390712833, 14063084452067724991010

Offset: 1

Seiichi Manyama, Jun 16 2019

Seiichi Manyama, Table of n, a(n) for n = 1..385

Diagonal of A308502.

Cf. A152211, A294956, A308594.

a[n_] := DivisorSum[n, #^(n/# + n) &]; Array[a, 20] (* Amiram Eldar, Mar 17 2021 *)

PARI
```
a(n) = sumdiv(n,d,d^(n/d+n));
```

my(N=20, x='x+O('x^N)); Vec(x*deriv(-log(prod(k=1, N, (1-k*(k*x)^k)^(1/k)))))

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(k+1)*x^k/(1-k^(k+1)*x^k))) \\ Seiichi Manyama, Mar 17 2021

from sympy import divisors
def A308668(n): return sum(d**(n//d+n) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

L.g.f.: -log(Product_{k>=1} (1 - k*(k*x)^k)^(1/k)) = Sum_{k>=1} a(k)*x^k/k.
G.f.: Sum_{k>=1} k^(k+1) * x^k/(1 - k^(k+1) * x^k). - Seiichi Manyama, Mar 17 2021
a(n) ~ n^(n+1). - Vaclav Kotesovec, Aug 30 2025

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A094471 a(n) = Sum_{(n - k)|n, 0 <= k <= n} k.

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A328203 Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.

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A309732 Expansion of Sum_{k>=1} k^2 * x^k/(1 - x^k)^3.

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A308668 a(n) = Sum_{d|n} d^(n/d+n).

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