A001159 -id:A001159 - cp's OEIS frontend

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

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

A066112 Numbers k such that sigma_4(k)/sigma_2(k) is an integer but not a prime.

Original entry on oeis.org

1, 16, 36, 48, 49, 64, 81, 100, 121, 144, 162, 180, 196, 225, 245, 256, 324, 361, 400, 432, 441, 484, 500, 529, 576, 605, 625, 648, 676, 729, 784, 841, 900, 931, 980, 1024, 1089, 1156, 1200, 1225, 1280, 1296, 1369, 1444, 1521, 1600, 1620, 1681, 1764, 1805

Offset: 1

Labos Elemer, Dec 06 2001

nonn

			The sequence includes squares, twice squares (such as 162 and 648), and other numbers (such as 48 and 180). The sigma_4/sigma_2 quotients usually have more than one distinct prime factor. Exception: sigma_4(48)/sigma_2(48) = 5732210/3410 = 1681 = 41^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A001157, A001159, A046871, A066109, A066110, A066111.

Do[s=DivisorSigma[4, n]; z=DivisorSigma[2, n]; If[IntegerQ[s/z]&&!PrimeQ[s/z], Print[n]], {n, 1, 10000}]

isok(k) = { my(f=sigma(k, 4)/sigma(k, 2)); !frac(f) && !isprime(f) } \\ Harry J. Smith, Feb 01 2010

Edited by Jon E. Schoenfield, Dec 24 2016

A297843 a(n) = Sum_{d|n} max(d, n/d)^4.

Original entry on oeis.org

1, 32, 162, 528, 1250, 2754, 4802, 8704, 13203, 21250, 29282, 44576, 57122, 81634, 102500, 139520, 167042, 225666, 260642, 341250, 393764, 497794, 559682, 715808, 781875, 971074, 1076004, 1310946, 1414562, 1743842, 1847042, 2236416, 2401124, 2839714, 3006052

Offset: 1

Seiichi Manyama, Jan 07 2018

nonn

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

Sum_{d|n} max(d, n/d)^k: A117003 (k=1), A297841 (k=2), A297842 (k=3), this sequence (k=4), A297844 (k=5).

Cf. A001159, A013663, A297794.

f[n_] := Block[{d = Divisors@ n}, Plus @@ (Max[#, n/#]^4 & /@ d)]; Array[f, 35] (* Robert G. Wilson v, Jan 07 2018 *)

PARI
```
{a(n) = sumdiv(n, d, max(d, n/d)^4)}
```

a(n) + A297794(n) = 2*A001159(n).

Sum_{k=1..n} a(k) ~ (2*zeta(5)/5) * n^5. - Amiram Eldar, Jan 12 2025

A300909 Sum of 4th powers dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 82, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17

Offset: 1

Ilya Gutkovskiy, Mar 15 2018

Multiplicative with a(p^e) = (p^(4*(1+floor(e/4)))-1)/(p^4-1). - Robert Israel, Mar 15 2018

			a(16) = 17 because 16 has 5 divisors {1, 2, 4, 8, 16} among which 2 divisors {1, 16} are 4th powers and 1 + 16 = 17.
L.g.f.: L(x) = x + x^2/2 + x^3/3 + x^4/4 + x^5/5 + x^6/6 + x^7/7 + x^8/8 + x^9/9 + x^10/10 + x^11/11 + x^12/12 + x^13/13 + x^14/14 + x^15/15 + 17*x^16/16 + x^17/17 + ...
exp(L(x)) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + 2*x^16 + 2*x^17 + ... + A046042(n)*x^n + ...

Antti Karttunen, Table of n, a(n) for n = 1..65537
A. Dixit, B. Maji, A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=4,k=4,n).

Cf. A000583, A001159, A035316, A046042, A046100 (positions of ones), A063775, A113061.

N:= 1000: # for a(1)..a(N)
V:= Vector(N,1):
for m from 2 to floor(N^(1/4)) do
  R:= [seq(i,i=m^4 .. N, m^4)];
  V[R]:= map(`+`,V[R],m^4)
od:
convert(V,list); # Robert Israel, Mar 15 2018

Table[DivisorSum[n, # &, IntegerQ[#^(1/4)] &], {n, 112}]
nmax = 112; Rest[CoefficientList[Series[Sum[k^4 x^k^4/(1 - x^k^4), {k, 1, 10}], {x, 0, nmax}], x]]
nmax = 112; Rest[CoefficientList[Series[-Log[Product[(1 - x^k^4), {k, 1, 10}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := (p^(4*(1 + Floor[e/4])) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

a(n) = sumdiv(n, d, d*ispower(d, 4)); \\ Michel Marcus, Mar 15 2018

G.f.: Sum_{k>=1} k^4*x^(k^4)/(1 - x^(k^4)).
L.g.f.: -log(Product_{k>=1} (1 - x^(k^4))) = Sum_{n>=1} a(n)*x^n/n.
D.g.f.: zeta(s)*zeta(4s-4). - Robert Israel, Mar 15 2018
Sum_{k=1..n} a(k) ~ zeta(5/4)*n^(5/4)/5 - n/2. - Vaclav Kotesovec, Dec 01 2020

A347142 Sum of 4th powers of divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 17, 1, 17, 1, 17, 1, 98, 1, 17, 82, 17, 1, 98, 1, 273, 82, 17, 1, 354, 1, 17, 82, 273, 1, 723, 1, 273, 82, 17, 626, 354, 1, 17, 82, 898, 1, 1394, 1, 273, 707, 17, 1, 1650, 1, 642, 82, 273, 1, 1394, 626, 2674, 82, 17, 1, 2275, 1, 17, 2483, 273, 626

Offset: 1

Ilya Gutkovskiy, Aug 19 2021

nonn

Cf. A001159, A056924, A070039, A279363, A339353, A339354, A347143.

Table[DivisorSum[n, #^4 &, # < Sqrt[n] &], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[k^4 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A347142(n) = { my(s=0); fordiv(n,d,if((d^2)>=n,return(s)); s += (d^4)); }; \\ Antti Karttunen, Aug 19 2021

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

A347143 Sum of 4th powers of divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 17, 1, 17, 1, 17, 82, 17, 1, 98, 1, 17, 82, 273, 1, 98, 1, 273, 82, 17, 1, 354, 626, 17, 82, 273, 1, 723, 1, 273, 82, 17, 626, 1650, 1, 17, 82, 898, 1, 1394, 1, 273, 707, 17, 1, 1650, 2402, 642, 82, 273, 1, 1394, 626, 2674, 82, 17, 1, 2275, 1, 17, 2483, 4369, 626

Offset: 1

Ilya Gutkovskiy, Aug 19 2021

nonn

Cf. A001159, A038548, A066839, A095118, A279363, A280375, A347142.

Table[DivisorSum[n, #^4 &, # <= Sqrt[n] &], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[k^4 x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A347143(n) = { my(s=0); fordiv(n,d,if((d^2)>n,return(s)); s += (d^4)); (s); }; \\ Antti Karttunen, Aug 19 2021

G.f.: Sum_{k>=1} k^4 * x^(k^2) / (1 - x^k).

A363608 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^5.

Original entry on oeis.org

0, 0, 0, 1, 5, 15, 35, 71, 126, 215, 330, 511, 715, 1036, 1370, 1891, 2380, 3201, 3876, 5061, 6020, 7645, 8855, 11207, 12655, 15665, 17676, 21512, 23751, 29000, 31465, 37851, 41250, 48756, 52400, 62602, 66045, 77691, 82966, 96521, 101270, 118966, 123410, 143397

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A013663, A069153, A363607.

Cf. A000203, A001157, A001158, A001159.

a[n_] := DivisorSum[n, Binomial[#, 4] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1-x^k)^5)))

a(n) = my(f = factor(n)); (sigma(f, 4) - 6*sigma(f, 3) + 11*sigma(f, 2) - 6*sigma(f)) / 24; \\ Amiram Eldar, Dec 30 2024

G.f.: Sum_{k>0} binomial(k,4) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d,4).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_4(n) - 6*sigma_3(n) + 11*sigma_2(n) - 6*sigma_1(n)) / 24.
Dirichlet g.f.: zeta(s) * (zeta(s-4) - 6*zeta(s-3) + 11*zeta(s-2) - 6*zeta(s-1)) / 24.
Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)

A046840 Numbers k such that the number of divisors of k divides the sum of the 4th powers of the divisors of k.

Original entry on oeis.org

1, 3, 4, 5, 7, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 37, 39, 41, 43, 44, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 100, 101, 103, 105, 107, 108, 109, 111

Offset: 1

Labos Elemer

nonn

A020486 is very similar to this sequence, but it does not include the following values below 1000 (which this sequence does include): {16, 80, 81, 176, 304, 324, 400, 405, 464, 496, 656, 784, 880, 891, 944, 976}.

			k = 16 is a term since it has 5 divisors, and sigma_4(16) = 69905 is divisible by 5.

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

Cf. A000005, A001159.

Cf. A003601, A020486, A046839.

[n: n in [1..120] | IsZero(DivisorSigma(4, n) mod NumberOfDivisors(n))]; // Bruno Berselli, Apr 11 2013

Select[Range[120], Divisible[DivisorSigma[4, #], DivisorSigma[0, #]] &] (* Amiram Eldar, Mar 17 2025 *)

isok(n) = sigma(n, 4) % numdiv(n) == 0; \\ Michel Marcus, May 13 2018

A236329 a(n) = sigma(n,1) * sigma(n,2) * ... * sigma(n,n).

Original entry on oeis.org

1, 15, 1120, 2929563, 38464354656, 80529415686720000, 538252697895729090560000, 1045011472134222568417452736171875, 14983270528936392555878952946810076508388237, 30023920804570215919584229032152609459437167079578240000

Offset: 1

Colin Barker, Jan 22 2014

nonn

sigma(n, k) is the sum of the k-th powers of the divisors of n.

			a(4) = sigma(4,1)*sigma(4,2)*sigma(4,3)*sigma(4,4) = 7*21*73*273 = 2929563.

Vaclav Kotesovec, Table of n, a(n) for n = 1..35

Cf. A236328.

Cf. A000203, A001157, A001158, A001159, A001160.

with(NumberTheory): seq(product(sigma[k](n), k = 1..n), n = 1..10); # Vaclav Kotesovec, Aug 20 2019

Table[Times@@DivisorSigma[Range[n],n],{n,10}] (* Harvey P. Dale, Oct 21 2017 *)

vector(12, n, prod(k=1, n, sigma(n, k)))

log(a(n)) ~ n*(n+1)*log(n)/2. - Vaclav Kotesovec, Aug 21 2019

A347158 Sum of 4th powers of distinct prime divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 16, 0, 16, 0, 16, 0, 97, 0, 16, 81, 16, 0, 97, 0, 16, 81, 16, 0, 97, 0, 16, 81, 16, 0, 722, 0, 16, 81, 16, 625, 97, 0, 16, 81, 641, 0, 97, 0, 16, 706, 16, 0, 97, 0, 641, 81, 16, 0, 97, 625, 2417, 81, 16, 0, 722, 0, 16, 2482, 16, 625, 97, 0, 16, 81, 3042

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Cf. A001159, A005065, A005068, A333806, A333808, A347142, A347156, A347157.

Table[DivisorSum[n, #^4 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 70}]
nmax = 70; CoefficientList[Series[Sum[Prime[k]^4 x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} prime(k)^4 * x^(prime(k)*(prime(k) + 1)) / (1 - x^prime(k)).

cp's OEIS Frontend

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

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A066112 Numbers k such that sigma_4(k)/sigma_2(k) is an integer but not a prime.

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A297843 a(n) = Sum_{d|n} max(d, n/d)^4.

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A300909 Sum of 4th powers dividing n.

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A347142 Sum of 4th powers of divisors of n that are < sqrt(n).

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A347143 Sum of 4th powers of divisors of n that are <= sqrt(n).

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A363608 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^5.

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A046840 Numbers k such that the number of divisors of k divides the sum of the 4th powers of the divisors of k.

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