A064603 -id:A064603 - cp's OEIS frontend

A064606 Numbers k such that A064603(k) is divisible by k.

1, 2, 7, 45, 184, 210, 267, 732, 1282, 3487, 98374, 137620, 159597, 645174, 3949726, 7867343, 13215333, 14153570, 14262845, 317186286, 337222295, 2788845412, 10937683400, 72836157215, 95250594634

Offset: 1

Labos Elemer, Sep 24 2001

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
a(22) > 2*10^9. - Donovan Johnson, Jun 21 2010
a(26) > 10^11, if it exists. - Amiram Eldar, Jan 18 2024

			Adding divisor-cube sums for j = 1..7 gives 1+9+28+73+126+252+344 = 833 = 7*119, which is divisible by 7, so 7 is a term and the integer quotient is 119.

Cf. A001158, A064603, A050226, A056650, A064605, A064607, A064610-A064612, A048290, A062982, A045345.

A064603 = Accumulate[Table[DivisorSigma[3, k], {k, 1, 1000000}]]; Select[Range[Length[A064603]], Divisible[A064603[[#]], #] &] (* Vaclav Kotesovec, Jul 11 2021 *)

(Sum_{j=1..k} sigma_3(j)) mod k = A064603(k) mod k = 0.

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

a(22)-a(25) from Amiram Eldar, Jan 18 2024

A001158 sigma_3(n): sum of cubes of divisors of n.

Original entry on oeis.org

1, 9, 28, 73, 126, 252, 344, 585, 757, 1134, 1332, 2044, 2198, 3096, 3528, 4681, 4914, 6813, 6860, 9198, 9632, 11988, 12168, 16380, 15751, 19782, 20440, 25112, 24390, 31752, 29792, 37449, 37296, 44226, 43344, 55261, 50654, 61740, 61544, 73710, 68922, 86688

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).
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..24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
Also the eigenvalues of the Hecke operator T_n for the entire modular normalized Eisenstein form E_4(z) (see A004009): T_n E_4 = a(n) E_4, n >= 1. For the Hecke operator T_n and eigenforms see, e.g., the Koecher-Krieg reference, p. 207, eq. (5) and p. 211, section 4, or the Apostol reference p. 120, eq. (13) and pp. 129 - 133. - Wolfdieter Lang, Jan 28 2016

			G.f. = x + 9*x^2 + 28*x^3 + 73*x^4 + 126*x^5 + 252*x^6 + 344*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 120, 129 - 133.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 166.
Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 207, 211.
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).
Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 17, G_4(z).

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, p. 827.
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
D. B. Lahiri, Some arithmetical identities for Ramanujan's and divisor functions, Bulletin of the Australian Mathematical Society, Volume 1, Issue 3 December 1969, pp. 307-314. See Theorem 2 p. 308.
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)

Cf. A000005, A000203, A001157.
Cf. A051731, A127093.
Cf. A027748, A124010.
Cf. A004009, A064603 (partial sums).

a001158 n = product $ zipWith (\p e -> (p^(3*e + 3) - 1) `div` (p^3 - 1))
                      (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jun 30 2013

[DivisorSigma(3,n): n in [1..40]]; // Bruno Berselli, Apr 10 2013

seq(numtheory:-sigma[3](n),n=1..100); # Robert Israel, Feb 05 2016

Table[DivisorSigma[3,n],{n,100}] (* corrected by T. D. Noe, Mar 22 2009 *)

makelist(divsum(n,3),n,1,100); /* Emanuele Munarini, Mar 26 2011 */

N=99; q='q+O('q^N);
Vec(sum(n=1,N,n^3*q^n/(1-q^n))) /* Joerg Arndt, Feb 04 2011 */

{a(n) = if( n<1, 0, sumdiv(n, d, d^3))}; /* Michael Somos, Jan 07 2017 */

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 3)
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Jan 09 2021

[sigma(n, 3) for n in range(1, 40)]  # Zerinvary Lajos, Jun 04 2009

Multiplicative with a(p^e) = (p^(3e+3)-1)/(p^3-1). - David W. Wilson, Aug 01 2001
Dirichlet g.f. zeta(s)*zeta(s-3). - R. J. Mathar, Mar 04 2011
G.f.: sum(k>=1, k^3*x^k/(1-x^k)). - Benoit Cloitre, Apr 21 2003
Equals A051731 * [1, 8, 27, 64, 125, ...] = A127093 * [1, 4, 9, 16, 25, ...]. - Gary W. Adamson, Nov 02 2007
L.g.f.: -log(Product_{j>=1} (1-x^j)^(j^2)) = (1/1)*z^1 + (9/2)*z^2 + (28/3)*z^3 + (73/4)*z^4 + ... + (a(n)/n)*z^n + ... - Joerg Arndt, Feb 04 2011
a(n) = Sum{d|n} tau_{-2}^d*J_3(n/d), where tau_{-2} is A007427 and J_3 is A059376. - Enrique Pérez Herrero, Jan 19 2013
a(n) = A004009(n)/240. - Artur Jasinski, Sep 06 2016. See, e.g., Hardy, p. 166, (10.5.6), with Q = E_4, and with present offset 0. - Wolfdieter Lang, Jan 31 2017
8*a(n) = sum of cubes of even divisors of 2*n. - Wolfdieter Lang, Jan 07 2017
G.f.: Sum_{n >= 1} x^n*(1 + 4*x^n + x^(2*n))/(1 - x^n)^4. - Peter Bala, Jan 11 2021
Faster converging g.f.: Sum_{n >= 1} q^(n^2)*( n^3 + ((n + 1)^3 - 3*n^3)*q^n + (4 - 6*n^2)*q^(2*n) + (3*n^3 - (n - 1)^3)*q^(3*n) - n^3*q^(4*n) )/(1 - q^n)^4 - apply the operator x*d/dx three times to equation 5 in Arndt and then set x = 1. - Peter Bala, Jan 21 2021
a(n) = Sum_{1 <= i, j, k <= n} tau(gcd(i, j, k, n)) = Sum_{d divides n} tau(d)* J_3(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 22 2024

A064602 Partial sums of A001157: Sum_{j=1..n} sigma_2(j).

Original entry on oeis.org

1, 6, 16, 37, 63, 113, 163, 248, 339, 469, 591, 801, 971, 1221, 1481, 1822, 2112, 2567, 2929, 3475, 3975, 4585, 5115, 5965, 6616, 7466, 8286, 9336, 10178, 11478, 12440, 13805, 15025, 16475, 17775, 19686, 21056, 22866, 24566, 26776, 28458, 30958

Offset: 1

Labos Elemer, Sep 24 2001

nonn

In general, for m >= 0 and j >= 0, Sum_{k=1..n} k^m * sigma_j(k) = Sum_{k=1..s} (k^m * F_{m+j}(floor(n/k)) + k^(m+j) * F_m(floor(n/k))) - F_{m+j}(s) * F_m(s), where s = floor(sqrt(n)) and F_m(x) are the Faulhaber polynomials defined as F_m(x) = (Bernoulli(m+1, x+1) - Bernoulli(m+1, 1)) / (m+1). - Daniel Suteu, Nov 27 2020

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Project Euler, Problem 401: Sum of squares of divisors, 2012.

Cf. A001157, A064605.

Cf. A064603, A064604, A248076.

Accumulate@ Array[DivisorSigma[2, #] &, 42] (* Michael De Vlieger, Jan 02 2017 *)

a(n) = sum(j=1, n, sigma(j, 2)); \\ Michel Marcus, Dec 15 2013

f(n) = n*(n+1)*(2*n+1)/6; \\ A000330
a(n) = my(s=sqrtint(n)); sum(k=1, s, f(n\k) + k^2*(n\k)) - s*f(s); \\ Daniel Suteu, Nov 26 2020

from math import isqrt
def f(n): return n*(n+1)*(2*n+1)//6
def a(n):
    s = isqrt(n)
    return sum(f(n//k) + k*k*(n//k) for k in range(1, s+1)) - s*f(s)
print([a(k) for k in range(1, 43)]) # Michael S. Branicky, Oct 01 2022 after Daniel Suteu

a(n) = a(n-1) + A001157(n) = Sum_{j=1..n} sigma_2(j) where sigma_2(j) = A001157(j).
a(n) = Sum_{i=1..n} i^2 * floor(n/i). - Enrique Pérez Herrero, Sep 15 2012
G.f.: (1/(1 - x))*Sum_{k>=1} k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 02 2017
a(n) ~ zeta(3) * n^3 / 3. - Vaclav Kotesovec, Sep 02 2018
a(n) = Sum_{k=1..s} (A000330(floor(n/k)) + k^2*floor(n/k)) - s*A000330(s), where s = floor(sqrt(n)). - Daniel Suteu, Nov 26 2020

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

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.

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).

A365439 a(n) = Sum_{k=1..n} binomial(floor(n/k)+4,5).

Original entry on oeis.org

1, 7, 23, 64, 135, 282, 493, 864, 1375, 2166, 3168, 4715, 6536, 9132, 12278, 16525, 21371, 27998, 35314, 44995, 55847, 69504, 84455, 103882, 124428, 150005, 177921, 212017, 247978, 292890, 339267, 395874, 455796, 526692, 600788, 691066, 782457, 891048, 1004814

Offset: 1

Seiichi Manyama, Oct 23 2023

Partial sums of A073570.

Cf. A006218, A024916, A064602, A064603, A064604, A364970, A365409.

a(n) = sum(k=1, n, binomial(n\k+4, 5));

from math import isqrt, comb
def A365439(n): return (-(s:=isqrt(n))**2*comb(s+4,4)+sum((q:=n//k)*(5*comb(k+3,4)+comb(q+4,4)) for k in range(1,s+1)))//5 # Chai Wah Wu, Oct 26 2023

a(n) = Sum_{k=1..n} binomial(k+3,4) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^5 = 1/(1-x) * Sum_{k>=1} binomial(k+3,4) * x^k/(1-x^k).
a(n) = (A064604(n)+6*A064603(n)+11*A064602(n)+6*A024916(n))/24. - Chai Wah Wu, Oct 26 2023

A366971 a(n) = Sum_{k=3..n} binomial(k,3) * floor(n/k).

Original entry on oeis.org

0, 0, 1, 5, 15, 36, 71, 131, 216, 346, 511, 756, 1042, 1441, 1907, 2527, 3207, 4128, 5097, 6371, 7737, 9442, 11213, 13538, 15848, 18734, 21744, 25423, 29077, 33743, 38238, 43818, 49440, 56104, 62694, 70979, 78749, 88154, 97580, 108790, 119450, 132680, 145021, 159974

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A363607.
Cf. A366968, A366969, A366970.
Cf. A024916, A064602, A064603, A366967.

a(n) = sum(k=3, n, binomial(k, 3)*(n\k));

from math import isqrt, comb
def A366971(n): return -comb((s:=isqrt(n))+1,4)*(s+1)+sum(comb((q:=n//w)+1,4)+(q+1)*comb(w,3) for w in range(1,s+1)) # Chai Wah Wu, Oct 30 2023

G.f.: 1/(1-x) * Sum_{k>=1} x^(3*k)/(1-x^k)^4 = 1/(1-x) * Sum_{k>=3} binomial(k,3) * x^k/(1-x^k).

a(n) = (A064603(n) - 3*A064602(n) + 2*A024916(n))/6. - Chai Wah Wu, Oct 30 2023

A248076 Partial sums of the sum of the 5th powers of the divisors of n: Sum_{i=1..n} sigma_5(i).

Original entry on oeis.org

1, 34, 278, 1335, 4461, 12513, 29321, 63146, 122439, 225597, 386649, 644557, 1015851, 1570515, 2333259, 3415660, 4835518, 6792187, 9268287, 12572469, 16673621, 21988337, 28424681, 36677981, 46446732, 58699434, 73107634, 90873690, 111384840, 136555392

Offset: 1

Wesley Ivan Hurt, Sep 30 2014

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

Cf. A001160 (sigma_5).
Cf. A024916: Partial sums of sigma(n)   = A000203(n).
Cf. A064602: Partial sums of sigma_2(n) = A001157(n).
Cf. A064603: Partial sums of sigma_3(n) = A001158(n).
Cf. A064604: Partial sums of sigma_4(n) = A001159(n).

[(&+[DivisorSigma(5,j): j in [1..n]]): n in [1..30]]; // G. C. Greubel, Nov 07 2018

with(numtheory): A248076:=n->add(sigma[5](i), i=1..n): seq(A248076(n), n=1..50);

Table[Sum[DivisorSigma[5, i], {i, n}], {n, 30}]
Accumulate[DivisorSigma[5, Range[30]]] (* Vaclav Kotesovec, Mar 30 2018 *)

lista(nn) = vector(nn, n, sum(i=1, n, sigma(i, 5))) \\ Michel Marcus, Sep 30 2014

from math import isqrt
def A248076(n): return ((s:=isqrt(n))**3*(s+1)**2*(1-2*s*(s+1)) + sum((q:=n//k)*(12*k**5+q*(q**2*(q*(2*q+6)+5)-1)) for k in range(1,s+1)))//12 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{i=1..n} sigma_5(i) = Sum_{i=1..n} A001160(i).
a(n) ~ Zeta(6) * n^6 / 6. - Vaclav Kotesovec, Sep 02 2018
a(n) ~ Pi^6 * n^6 / 5670. - Vaclav Kotesovec, Sep 02 2018
a(n) = Sum_{k=1..n} (Bernoulli(6, floor(1 + n/k)) - 1/42)/6, where Bernoulli(n,x) are the Bernoulli polynomials. - Daniel Suteu, Nov 07 2018
a(n) = Sum_{k=1..n} k^5 * floor(n/k). - Daniel Suteu, Nov 08 2018

cp's OEIS Frontend

A064606 Numbers k such that A064603(k) is divisible by k.

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A001158 sigma_3(n): sum of cubes of divisors of n.

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A064602 Partial sums of A001157: Sum_{j=1..n} sigma_2(j).

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PARI

Python

Formula

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

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Magma

Mathematica

PARI

Python

Formula

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

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Magma

Mathematica

PARI

Python

Formula

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

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Mathematica

PARI

PARI

Python