A064602 -id:A064602 - cp's OEIS frontend

A064605 Numbers k such that A064602(k) is divisible by k.

1, 2, 8, 74, 146, 150, 158, 307, 526, 541, 16157, 20289, 271343, 953614, 1002122, 2233204, 3015123, 15988923, 48033767, 85110518238

Offset: 1

Labos Elemer, Sep 24 2001

Analogous sequences for various arithmetical functions are A050226, A056650, A064605, A064606, A064607, A064610, A064611, A048290, A062982, A045345.
a(20) > 3*10^10. - Donovan Johnson, Aug 31 2012
a(21) > 10^11, if it exists. - Amiram Eldar, Jan 18 2024

			Summing divisor-square sums for j = 1..8 gives 1+5+10+21+26+50+50+85 = 248, which is divisible by 8, so 8 is a term and the integer quotient is 31.

Cf. A001157, A064602, A050226, A056650, A064606, A064607, A064610, A064611, A064612, A048290, A062982, A045345.

k = 1; lst = {}; s = 0; While[k < 1000000001, s = s + DivisorSigma[2, k]; If[ Mod[s, k] == 0, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G. Wilson v, Apr 25 2011 *)

(Sum_{j=1..k} sigma_2(j)) mod k = A064602(k) mod k = 0.

a(15)-a(19) from Donovan Johnson, Jun 21 2010

a(20) from Amiram Eldar, Jan 18 2024

A001157 a(n) = sigma_2(n): sum of squares of divisors of n.

Original entry on oeis.org

1, 5, 10, 21, 26, 50, 50, 85, 91, 130, 122, 210, 170, 250, 260, 341, 290, 455, 362, 546, 500, 610, 530, 850, 651, 850, 820, 1050, 842, 1300, 962, 1365, 1220, 1450, 1300, 1911, 1370, 1810, 1700, 2210, 1682, 2500, 1850, 2562, 2366, 2650, 2210, 3410, 2451, 3255

Offset: 1

N. J. A. Sloane, R. K. Guy

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
sigma_2(n) is the sum of the squares of the divisors of n.
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001.
Row sums of triangles A134575 and A134559. - Gary W. Adamson, Nov 02 2007
Also sum of square divisors of n^2. - Michel Marcus, Jan 14 2014
Conjecture: For each k = 2,3,..., all the rational numbers sigma_k(n)/n^k = Sum_{d|n} 1/d^k (n = 1,2,3,...) have pairwise distinct fractional parts. - Zhi-Wei Sun, Oct 15 2015
5 is the only prime entry in the sequence. - Drake Thomas, Dec 18 2016
4*a(n) = sum of squares of even divisors of 2*n. - Wolfdieter Lang, Jan 07 2017

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 827.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 11.
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table I. The entry 53 should be 50. - N. J. A. Sloane, May 21 2014
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See (2.3).
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Divisor Function
Index entries for sequences related to sigma(n)
Index entries for "core" sequences

Cf. A000005, A000203, A001158, A001159, A053807, A064602, A127093, A134841, A048250.
Cf. also A192794, A082063 (gcd(a(n),n) and its largest prime factor); A179931, A192795 (gcd(a(n),A000203(n)) and largest prime factor).
Main diagonal of the array in A242639.
Cf. A333972 (Dgf at s=4).

a001157 n = s n 1 1 a000040_list where
   s 1 1 y _          = y
   s m x y ps'@(p:ps)
     | m `mod` p == 0 = s (m `div` p) (x * p^2) y ps'
     | x > 1          = s m 1 (y * (x * p^2 - 1) `div` (p^2 - 1)) ps
     | otherwise      = s m 1 y ps
-- Reinhard Zumkeller, Jul 10 2011

[DivisorSigma(2,n): n in [1..50]]; // Bruno Berselli, Apr 10 2013

with(numtheory); A001157 := n->sigma[2](n); [seq(sigma[2](n), n=1..100)];

Table[DivisorSigma[2, n], {n, 1, 50}] (* Stefan Steinerberger, Mar 24 2006 *)
DivisorSigma[2,Range[50]] (* Harvey P. Dale, Aug 22 2016 *)

makelist(divsum(n,2),n,1,20); /* Emanuele Munarini, Mar 26 2011 */

PARI
```
a(n)=if(n<1,0,sigma(n,2))
```

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1-p^2*X))[n])

a(n)=if(n<1,0,n*polcoeff(sum(k=1,n,x^k/(x^k-1)^2/k,x*O(x^n)),n)) /* Michael Somos, Jan 29 2005 */

N=99; q='q+O('q^N); Vec(sum(n=1,N,n^2*q^n/(1-q^n)))  /* Joerg Arndt, Feb 04 2011 */

a(n) = sumdiv(n^2, d, issquare(d)*d); \\ Michel Marcus, Jan 14 2014

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 2)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Jan 05 2021

from math import prod
from sympy import factorint
def a(n): return prod((p**(2*e+2)-1)//(p**2-1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Feb 25 2024

[sigma(n,2)for n in range(1,51)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>0} k^2 x^k/(1-x^k). Dirichlet g.f.: zeta(s)*zeta(s-2). - Michael Somos, Apr 05 2003
Multiplicative with a(p^e) = (p^(2e+2)-1)/(p^2-1). - David W. Wilson, Aug 01 2001
G.f. for sigma_k(n): Sum_{m>0} m^k*x^m/(1-x^m). - Vladeta Jovovic, Oct 18 2002
L.g.f.: -log(Product_{j>=1} (1-x^j)^j) = Sum_{n>=1} a(n)/n*x^n. - Joerg Arndt, Feb 04 2011
Equals A127093 * [1, 2, 3, ...]. - Gary W. Adamson, May 10 2007
Equals A051731 * [1, 4, 9, 16, 25, ...]. A051731 * [1/1, 1/2, 1/3, 1/4, ...] = [1/1, 5/4, 10/9, 21/16, 26/25, ...]. - Gary W. Adamson, Nov 02 2007
Row sums of triangle A134841. - Gary W. Adamson, Nov 12 2007
a(n) = A035316(n^2). - Michel Marcus, Jan 14 2014
Conjecture: a(n) = sigma(n^2*rad(n))/sigma(rad(n)), where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 20 2017
G.f.: Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
a(n) = a(n/4) + A050461(n) + A076577(n/2) + A050465(n) where A(.) are zero for non-integer arguments. - R. J. Mathar, May 25 2020
Sum_{k>=1} 1/a(k) = A109694 = 1.53781289182725616253866100273826833091936004947322354929617689659426330445... - Vaclav Kotesovec, Sep 26 2020
G.f.: Sum_{n >= 1} q^(n^2)*(n^2 - ((n-1)^2 - 2)*q^n - ((n+1)^2 - 2)*q^(2*n) + n^2*q^(3*n))/(1 - q^n)^3 - apply the operator x*d/dx twice to equation 5 in Arndt and set x = 1. - Peter Bala, Jan 21 2021
From Vaclav Kotesovec, Aug 07 2022: (Start)
Sum_{k=1..n} a(k) = A064602(n) ~ zeta(3) * n^3 / 3.
Sum_{k=1..n} (-1)^k * a(k) ~ zeta(3) * n^3 / 24. (End)
a(n) = Sum_{1 <= i, j <= n} tau(gcd(i, j, n)) = Sum_{d divides n} tau(d) * J_2(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 22 2024

A064603 Partial sums of A001158: Sum_{j=1..n} sigma_3(j).

Original entry on oeis.org

1, 10, 38, 111, 237, 489, 833, 1418, 2175, 3309, 4641, 6685, 8883, 11979, 15507, 20188, 25102, 31915, 38775, 47973, 57605, 69593, 81761, 98141, 113892, 133674, 154114, 179226, 203616, 235368, 265160, 302609, 339905, 384131, 427475, 482736

Offset: 1

Labos Elemer, Sep 24 2001

nonn

In general, Sum_{k=1..n} sigma_m(k) = Sum_{k=1..n} k^m * floor(n/k). - Daniel Suteu, Nov 08 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A001158, A064606.

Cf. A064602, A064604, A248076.

Accumulate@ Array[DivisorSigma[3, #] &, 36] (* Michael De Vlieger, Nov 03 2017 *)

a(n) = sum(j=1, n, sigma(j, 3)); \\ Michel Marcus, Nov 04 2017

a(n) = sum(k=1, n, k^3 * (n\k)); \\ Daniel Suteu, Nov 08 2018

from math import isqrt
def A064603(n): return (-(s:=isqrt(n))**3*(s+1)**2 + sum((q:=n//k)*(4*k**3+q*(q*(q+2)+1)) for k in range(1,s+1)))>>2 # Chai Wah Wu, Oct 21 2023

a(n) = a(n-1) + A001158(n) = Sum_{j=1..n} sigma_3(j), where sigma_3(j) = A001158(j).
G.f.: (1/(1 - x))*Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 23 2017
a(n) ~ Pi^4 * n^4 / 360. - Vaclav Kotesovec, Sep 02 2018
a(n) = Sum_{k=1..n} ((1/2) * floor(n/k) * floor(1 + n/k))^2. - Daniel Suteu, Nov 07 2018
a(n) = Sum_{k=1..n} k^3 * floor(n/k). - Daniel Suteu, Nov 08 2018

A318742 a(n) = Sum_{k=1..n} floor(n/k)^3.

Original entry on oeis.org

1, 9, 29, 74, 136, 254, 382, 596, 833, 1173, 1505, 2057, 2527, 3209, 3921, 4856, 5674, 6928, 7956, 9474, 10882, 12608, 14128, 16506, 18369, 20797, 23141, 26129, 28567, 32259, 35051, 38963, 42483, 46675, 50435, 55904, 59902, 65156, 70092, 76460

Offset: 1

Vaclav Kotesovec, Sep 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Richard J. Mathar, Chebyshev approximation of x^m (-log x)^l in the interval 0 <= x <= 1, arXiv:2408.15212 [math.CA], 2024. See p. 2.

Cf. A000005, A000203, A024916, A064602, A064603, A064604.

Cf. A006218, A222548, A318743, A318744.

[&+[Floor(n/k)^3:k in [1..n]]: n in [1..40]]; // Marius A. Burtea, Jul 16 2019

Table[Sum[Floor[n/k]^3, {k, 1, n}], {n, 1, 40}]
Accumulate[Table[DivisorSigma[0, k] - 3*DivisorSigma[1, k] + 3*DivisorSigma[2, k], {k, 1, 40}]]

a(n) = sum(k=1, n, (n\k)^3); \\ Michel Marcus, Sep 03 2018

from math import isqrt
def A318742(n): return -(s:=isqrt(n))**4 + sum((q:=n//k)*(3*k*(k-1)+q**2+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 21 2023

a(n) = A006218(n) - 3*A024916(n) + 3*A064602(n).
a(n) ~ zeta(3) * n^3.
G.f.: (1/(1 - x)) * Sum_{k>=1} (3*k*(k - 1) + 1) * x^k/(1 - x^k). - Ilya Gutkovskiy, Jul 16 2019

A064612 Partial sum of bigomega is divisible by n, where bigomega(n)=A001222(n) and summatory-bigomega(n)=A022559(n).

Original entry on oeis.org

1, 4, 5, 2178, 416417176, 416417184, 416417185, 416417186, 416417194, 416417204, 416417206, 416417208, 416417213, 416417214, 416417231, 416417271, 416417318, 416417319, 416417326, 416417335, 416417336, 416417338, 416417339, 416417374

Offset: 1

Labos Elemer, Sep 24 2001

nonn

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
Partial sums of A001222, similarly to summatory A001221 increases like loglog(n), explaining small quotients.
a(25) > 10^13. - Giovanni Resta, Apr 25 2017

			Sum of bigomega values from 1 to 5 is: 0+0+1+1+2+1=5, which is divisible by n=5, so 5 is here, with quotient=1. For the last value,2178,below 1000000 the quotient is only 3.

A001222, A022559, A050226, A056650, A064602-A064611, A048290, A062982, A045345.

Mod[A022559(n), n]=0

a(5)-a(24) from Donovan Johnson, Nov 15 2009

A318743 a(n) = Sum_{k=1..n} floor(n/k)^4.

Original entry on oeis.org

1, 17, 83, 274, 644, 1396, 2502, 4388, 6919, 10743, 15385, 22407, 30233, 41209, 53853, 70650, 88636, 113308, 138654, 172332, 207984, 252416, 298002, 358654, 417873, 492065, 569061, 663427, 756053, 875541, 989063, 1130915, 1272967, 1441383, 1607147, 1817080

Offset: 1

Vaclav Kotesovec, Sep 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A024916, A064602, A064603, A064604.

Cf. A006218, A222548, A318742, A318744.

[&+[Floor(n/k)^4:k in [1..n]]:n in [1..36]]; // Marius A. Burtea, Jul 16 2019

Table[Sum[Floor[n/k]^4, {k, 1, n}], {n, 1, 40}]
Accumulate[Table[-DivisorSigma[0, k] + 4*DivisorSigma[1, k] - 6*DivisorSigma[2, k] + 4*DivisorSigma[3, k], {k, 1, 40}]]

a(n) = sum(k=1, n, (n\k)^4); \\ Michel Marcus, Sep 03 2018

from math import isqrt
def A318743(n): return -(s:=isqrt(n))**5+sum((q:=n//k)*(k**4-(k-1)**4+q**3) for k in range(1,s+1)) # Chai Wah Wu, Oct 26 2023

a(n) = -A006218(n) + 4*A024916(n) - 6*A064602(n) + 4*A064603(n).
a(n) ~ zeta(4) * n^4.
a(n) ~ Pi^4 * n^4 / 90.
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * (2*k^2 - 2*k + 1) * x^k/(1 - x^k). - Ilya Gutkovskiy, Jul 16 2019

A318744 a(n) = Sum_{k=1..n} floor(n/k)^5.

Original entry on oeis.org

1, 33, 245, 1058, 3160, 8054, 17086, 33860, 60353, 103437, 164489, 257945, 380407, 556001, 779865, 1085840, 1457122, 1958008, 2544540, 3312306, 4205650, 5336264, 6618976, 8254674, 10059777, 12298021, 14792045, 17829881, 21130663, 25189011, 29518163, 34749419

Offset: 1

Vaclav Kotesovec, Sep 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A024916, A064602, A064603, A064604.

Cf. A006218, A222548, A318742, A318743.

Table[Sum[Floor[n/k]^5, {k, 1, n}], {n, 1, 40}]
Accumulate[Table[DivisorSigma[0, k] - 5*DivisorSigma[1, k] + 10*DivisorSigma[2, k] - 10*DivisorSigma[3, k] + 5*DivisorSigma[4, k], {k, 1, 40}]]

a(n) = sum(k=1, n, (n\k)^5); \\ Michel Marcus, Sep 03 2018

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^5-(k-1)^5)*x^k/(1-x^k))/(1-x)) \\ Seiichi Manyama, May 27 2021

from math import isqrt
def A318744(n): return -(s:=isqrt(n))**6+sum((q:=n//k)*(k**5-(k-1)**5+q**4) for k in range(1,s+1)) # Chai Wah Wu, Oct 26 2023

a(n) = A006218(n) - 5*A024916(n) + 10*A064602(n) - 10*A064603(n) + 5*A064604(n).

a(n) ~ zeta(5) * n^5.

A356298 a(n) = n! * Sum_{k=1..n} sigma_2(k)/k.

Original entry on oeis.org

1, 7, 41, 290, 2074, 18444, 165108, 1749264, 19412496, 241299360, 3097006560, 45546606720, 673536159360, 10986261431040, 187460277177600, 3445281394329600, 64637392771123200, 1325310849663897600, 27498565425087590400, 616389533324974080000

Offset: 1

Seiichi Manyama, Aug 03 2022

nonn

Cf. A001157, A064602, A356010, A356297, A356323.

Table[n! * Sum[DivisorSigma[2, k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)

PARI
```
a(n) = n!*sum(k=1, n, sigma(k, 2)/k);
```

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k*(1-x^k)^2))/(1-x)))

my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, k*log(1-x^k))/(1-x)))

E.g.f.: (1/(1-x)) * Sum_{k>0} x^k/(k * (1 - x^k)^2).
E.g.f.: -(1/(1-x)) * Sum_{k>0} k * log(1 - x^k).
a(n) ~ n! * zeta(3) * n^2 / 2. - Vaclav Kotesovec, Aug 07 2022

A319649 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} j^k * floor(n/j).

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 6, 8, 8, 1, 10, 16, 15, 10, 1, 18, 38, 37, 21, 14, 1, 34, 100, 111, 63, 33, 16, 1, 66, 278, 373, 237, 113, 41, 20, 1, 130, 796, 1335, 999, 489, 163, 56, 23, 1, 258, 2318, 4957, 4461, 2393, 833, 248, 69, 27, 1, 514, 6820, 18831, 20583, 12513, 4795, 1418, 339, 87, 29

Offset: 1

Ilya Gutkovskiy, Dec 09 2018

			Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   3,   4,    6,   10,    18,     34,  ...
   5,   8,   16,   38,   100,    278,  ...
   8,  15,   37,  111,   373,   1335,  ...
  10,  21,   63,  237,   999,   4461,  ...
  14,  33,  113,  489,  2393,  12513,  ...

Index entries for sequences related to sigma(n)

Columns k=0..5 give A006218, A024916, A064602, A064603, A064604, A248076.

Cf. A082771, A109974, A319194 (diagonal).

Table[Function[k, Sum[j^k Floor[n/j] , {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[DivisorSigma[k, j], {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

from itertools import count, islice
from math import isqrt
from sympy import bernoulli
def A319649_T(n,k): return (((s:=isqrt(n))+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))+sum(w**k*(k+1)*((q:=n//w)+1)-bernoulli(k+1)+bernoulli(k+1,q+1) for w in range(1,s+1)))//(k+1) + int(k==0)
def A319649_gen(): # generator of terms
     return (A319649_T(k+1,n-k-1) for n in count(1) for k in range(n))
A319649_list = list(islice(A319649_gen(),30)) # Chai Wah Wu, Oct 24 2023

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} j^k*x^j/(1 - x^j).

A(n,k) = Sum_{j=1..n} sigma_k(j).

A350123 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^2.

Original entry on oeis.org

1, 8, 22, 57, 91, 185, 247, 402, 545, 775, 917, 1379, 1573, 1995, 2455, 3106, 3428, 4377, 4775, 5909, 6753, 7727, 8301, 10331, 11230, 12564, 13904, 15990, 16888, 19908, 20930, 23597, 25545, 27767, 29827, 34468, 35910, 38660, 41328, 46318, 48080, 53644, 55578

Offset: 1

Seiichi Manyama, Dec 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A064602, A143128, A222548, A350107, A350124, A350125, A350128.

Accumulate[Table[2*k*DivisorSigma[1, k] - DivisorSigma[2, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 16 2021 *)

PARI
```
a(n) = sum(k=1, n, k^2*(n\k)^2);
```

a(n) = sum(k=1, n, k^2*sumdiv(k, d, (2*d-1)/d^2));

a(n) = sum(k=1, n, 2*k*sigma(k)-sigma(k, 2));

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k*(1+x^k)/(1-x^k)^3)/(1-x))

from math import isqrt
def A350123(n): return (-(s:=isqrt(n))**3*(s+1)*((s<<1)+1)+sum((q:=n//k)*(6*k**2*q+((k<<1)-1)*(q+1)*((q<<1)+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 24 2023

a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (2*d - 1)/d^2 = Sum_{k=1..n} 2 * k * sigma(k) - sigma_2(k) = 2 * A143128(n) - A064602(n).
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k * (1 + x^k)/(1 - x^k)^3.
a(n) ~ n^3 * (Pi^2/9 - zeta(3)/3). - Vaclav Kotesovec, Dec 16 2021

cp's OEIS Frontend

A064605 Numbers k such that A064602(k) is divisible by k.

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A001157 a(n) = sigma_2(n): sum of squares of divisors of n.

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A064603 Partial sums of A001158: Sum_{j=1..n} sigma_3(j).

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A318742 a(n) = Sum_{k=1..n} floor(n/k)^3.

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A064612 Partial sum of bigomega is divisible by n, where bigomega(n)=A001222(n) and summatory-bigomega(n)=A022559(n).

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A318743 a(n) = Sum_{k=1..n} floor(n/k)^4.

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