A064604 -id:A064604 - cp's OEIS frontend

A064607 Numbers k such that A064604(k) is divisible by k.

1, 2, 7, 151, 257, 1823, 3048, 5588, 6875, 7201, 8973, 24099, 5249801, 9177919, 18926164, 70079434, 78647747, 705686794, 2530414370, 3557744074, 25364328389, 32487653727, 66843959963

Offset: 1

Labos Elemer, Sep 24 2001

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
a(19) > 2*10^9. - Donovan Johnson, Jun 21 2010
a(24) > 10^11, if it exists. - Amiram Eldar, Jan 18 2024

			Adding 4th-power divisor-sums for j = 1..7 gives 1+17+82+273+626+1394+2402 = 4795 which is divisible by 7, so 7 is a term and the integer quotient is 655.

Cf. A001159, A064604, A050226, A056650, A064605, A064606, A064610-A064612, A048290, A062982, A045345.

k = 1; lst = {}; s = 0; While[k < 1000000001, s = s + DivisorSigma[4, k]; If[ Mod[s, k] == 0, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G.Wilson v, Aug 25 2011 *)

(Sum_{j=1..k} sigma_4(j)) mod k = A064604(k) mod k = 0.

a(13)-a(18) from Donovan Johnson, Jun 21 2010

a(19)-a(23) from Amiram Eldar, Jan 18 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

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

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

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

A355887 a(n) = Sum_{k=1..n} k^k * floor(n/k).

Original entry on oeis.org

1, 6, 34, 295, 3421, 50109, 873653, 17651130, 405071647, 10405074777, 295716745389, 9211817240589, 312086923832843, 11424093750214407, 449317984131076935, 18896062057857406028, 846136323944194170206, 40192544399241119212807

Offset: 1

Seiichi Manyama, Jul 20 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Partial sums of A062796.

Cf. A006218, A024916, A064602, A064603, A064604, A248076, A319194.

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

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

def A355887(n): return n*(1+n**(n-1))+sum(k**k*(n//k) for k in range(2,n)) if n>1 else 1 # Chai Wah Wu, Jul 21 2022

a(n) = Sum_{k=1..n} Sum_{d|k} d^d.

G.f.: (1/(1-x)) * Sum_{k>0} (k * x)^k/(1 - x^k).

cp's OEIS Frontend

A064607 Numbers k such that A064604(k) is divisible by k.

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

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

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

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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).

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