A001159 -id:A001159 - cp's OEIS frontend

A082771 Triangular array, read by rows: t(n,k) = Sum_{d|n} d^k, 0 <= k < n.

1, 2, 3, 2, 4, 10, 3, 7, 21, 73, 2, 6, 26, 126, 626, 4, 12, 50, 252, 1394, 8052, 2, 8, 50, 344, 2402, 16808, 117650, 4, 15, 85, 585, 4369, 33825, 266305, 2113665, 3, 13, 91, 757, 6643, 59293, 532171, 4785157, 43053283, 4, 18, 130, 1134, 10642, 103158, 1015690, 10078254, 100390882, 1001953638

Offset: 1

Reinhard Zumkeller, May 21 2003

			From _R. J. Mathar_, Dec 06 2006 (Start):
The triangle may be extended to a rectangular array (A319278):
  1  1   1    1     1 1 1 1 1 1 1 ...
  2  3   5    9    17 33 65 129 257 513 1025 ...
  2  4  10   28    82 244 730 2188 6562 19684 59050 ...
  3  7  21   73   273 1057 4161 16513 65793 262657 1049601 ...
  2  6  26  126   626 3126 15626 78126 390626 1953126 9765626 ...
  4 12  50  252  1394 8052 47450 282252 1686434 10097892 60526250 ...
  2  8  50  344  2402 16808 117650 823544 5764802 40353608 282475250 ...
  4 15  85  585  4369 33825 266305 2113665 16843009 134480385 1074791425 ...
  3 13  91  757  6643 59293 532171 4785157 43053283 387440173 3486843451 ...
  4 18 130 1134 10642 103158 1015690 10078254 100390882 1001953638... (End)

T. D. Noe, Rows n=1..100, flattened
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000005, A000203, A001157, A001158, A001159, A001160.
Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963.
Cf. A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.

T:= (n,k)-> numtheory[sigma][k](n):
seq(seq(T(n,k), k=0..n-1), n=1..10);  # Alois P. Heinz, Oct 25 2024

T[n_, k_] := DivisorSigma[k, n];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Dec 16 2021 *)

row(n) = {my(f = factor(n)); vector(n, k, sigma(f, k-1));} \\ Amiram Eldar, May 09 2025

t(n, k) = Product(((p^((e(n, p)+1)*k))-1)/(p^k-1): n=Product(p^e(n, p): p prime)), 0<=k

t(n,0) = A000005(n), t(n,n) = A023887(n).

t(n,1) = A000203(n), n>1; t(n,2) = A001157(n), n>2; t(n,3) = A001158(n), n>3.

t(n,4) = A001159(n), n>4; t(n,5) = A001160(n), n>5; t(n,6) = A013954(n), n>6.

From R. J. Mathar, Oct 29 2006: (Start)

t(2,k) = A000051(k); t(3,k) = A034472(k); t(4,k) = A001576(k);

t(5,k) = A034474(k); t(6,k) = A034488(k); t(7,k) = A034491(k);

t(8,k) = A034496(k); t(9,k) = A034513(k); t(10,k) = A034517(k);

t(11,k) = A034524(k); t(12,k) = A034660(k). (End)

Corrected by R. J. Mathar, Dec 05 2006

A279395 a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^4.

Original entry on oeis.org

1, 15, 82, 271, 626, 1230, 2402, 4367, 6643, 9390, 14642, 22222, 28562, 36030, 51332, 69903, 83522, 99645, 130322, 169646, 196964, 219630, 279842, 358094, 391251, 428430, 538084, 650942, 707282, 769980, 923522, 1118479, 1200644, 1252830, 1503652, 1800253, 1874162, 1954830, 2342084, 2733742

Offset: 1

Wolfdieter Lang, Jan 09 2017

This is the k=4 member of the family sigma^*_k(n), defined in the Hardy reference, which is sigma_k(2*j+1) if n = 2*j+1 and sigma_k^e(2*j) - sigma_k^o(2*j) if n=2*j, where the superscript e and o stands for a restriction to even and odd divisors in the sum of their k-th powers, respectively.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher

Cf. A112329 (k=0), A113184 (k=1), A064027 (k=2), A008457(k=3).

[&+[(-1)^(n-d)*d^4:d in Divisors(n)]:n in [1..40]]; // Marius A. Burtea, Aug 17 2019

# A version with signs - N. J. A. Sloane, Nov 23 2018
zet1:=(n,i)->add((-1)^(d-1)*d^i, d in divisors(n));
szet1:=i->[seq(zet1(n,i),n=1..120)];
szet1(4);

f[p_, e_] := If[p == 2, (2^(4*(e + 1)) - 31)/15, (p^(4*(e + 1)) - 1)/(p^4 - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 40] (* Amiram Eldar, Aug 17 2019 *)

a(n) = sumdiv(n, d, (-1)^(n-d)*d^4); \\ Michel Marcus, Jan 09 2017

a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^4.
Bisection: a(2*j-1) = A001159(2*j-1), a(2*j) = 16*A001159(j) - A051001(j), j >= 1. See the comment above for k=4, and the Hardy reference.
G.f.: Sum_{k>=1} k^4*x^k/(1-(-x)^k).
Multiplicative with a(2^k) = 2^4*(2^(4*k)-1)/(2^4-1) - 1 = (2^(4*(k+1)) - 31)/15 and a(p^k) = (p^(4*(k+1))-1)/(p^4-1) for primes p > 2 (see A001159).

A301548 Expansion of Product_{k>=1} (1 + x^k)^(sigma_4(k)).

Original entry on oeis.org

1, 1, 17, 99, 491, 2429, 12056, 56618, 259074, 1155193, 5044288, 21585280, 90694483, 374661505, 1524090522, 6111565745, 24181962002, 94491963120, 364920615165, 1393789672170, 5268145436728, 19715988877445, 73096492576283, 268589397735778, 978533798885874

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..3000

Cf. A001159, A107742, A301542.

nmax = 40; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[4, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(6^(2/3) * Pi * (31*Zeta(5)/7)^(1/6) * n^(5/6)/5 + Pi *(7/(31*Zeta(5)))^(1/6) * n^(1/6) / (240*6^(2/3))) * (31*Zeta(5)/7)^(1/12) / (2^(7/6) * 3^(2/3) * n^(7/12)).

G.f.: exp(Sum_{k>=1} sigma_5(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 26 2018

A386748 a(n) = n^3*sigma_4(n).

Original entry on oeis.org

0, 1, 136, 2214, 17472, 78250, 301104, 823886, 2236928, 4842747, 10642000, 19488502, 38683008, 62750714, 112048496, 173245500, 286330880, 410343586, 658613592, 893878598, 1367184000, 1824083604, 2650436272, 3404837614, 4952558592, 6113296875, 8534097104, 10591107372

Offset: 0

Vaclav Kotesovec, Aug 01 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..8000

Cf. A000578, A001159, A386749, A386747.

[0] cat [n^3*DivisorSigma(4, n): n in [1..35]]; // Vincenzo Librandi, Aug 02 2025

Table[n^3*DivisorSigma[4, n], {n, 0, 40}]
nmax = 40; CoefficientList[Series[Sum[k^7*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=1} k^7*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4. - Amiram Eldar, Aug 01 2025
a(n) = n^3*A001159(n).
Dirichlet g.f.: zeta(s-3)*zeta(s-7). - R. J. Mathar, Aug 03 2025

A046871 Numbers k such that sigma_2(k) divides sigma_4(k).

Original entry on oeis.org

1, 4, 9, 16, 20, 25, 36, 48, 49, 64, 81, 100, 121, 144, 162, 169, 180, 196, 225, 245, 256, 289, 324, 361, 400, 432, 441, 484, 500, 529, 576, 605, 625, 648, 676, 729, 784, 841, 900, 931, 961, 980, 1024, 1089, 1156, 1200, 1225, 1280, 1296, 1369, 1444, 1521

Offset: 1

Labos Elemer

nonn

sigma_2(k) is the sum of the squares of the divisors of k (A001157).

sigma_4(k) is the sum of the 4th powers of the divisors of k (A001159).

			k = a(8) = 48 of which divisor power sums for powers 0, 1, 2, 3, 4 are 10, 124, 3410, 131068, 5732210, respectively. Here sigma_2(k) = 3410 and sigma_4(k) = 3410*1681.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)

Cf. A001157, A001159, A020487, A020486, A046841.

[n: n in [1..1600] | IsZero(DivisorSigma(4,n) mod DivisorSigma(2,n))]; // Bruno Berselli, Apr 10 2013

Select[Range@ 1600, Divisible[DivisorSigma[4, #], DivisorSigma[2, #]] &] (* Michael De Vlieger, May 20 2017 *)

isok(n) = !(sigma(n, 4) % sigma(n, 2)); \\ Michel Marcus, May 21 2017

A065827 Sum of squares of divisors of square numbers.

Original entry on oeis.org

1, 21, 91, 341, 651, 1911, 2451, 5461, 7381, 13671, 14763, 31031, 28731, 51471, 59241, 87381, 83811, 155001, 130683, 221991, 223041, 310023, 280371, 496951, 406901, 603351, 597871, 835791, 708123, 1244061, 924483, 1398101, 1343433, 1760031, 1595601, 2516921

Offset: 1

Vladeta Jovovic, Dec 06 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Harry J. Smith)

Cf. A001157, A065764.

A065827 := proc(n) numtheory[sigma][2](n^2) ; end proc:
seq(A065827(n),n=1..20) ; # R. J. Mathar, Apr 01 2011

DivisorSigma[2,#]&/@(Range[40]^2) (* Harvey P. Dale, May 18 2011 *)
f[p_, e_] := (p^(4*e + 2) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* Amiram Eldar, Sep 13 2020 *)

{ for (n=1, 500, a=sigma(n^2, 2); write("b065827.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 01 2009

[sigma(n^2,2)for n in range(1,34)] # Zerinvary Lajos, Jun 13 2009

Multiplicative with a(p^e) = (p^(4*e+2)-1)/(p^2-1).
a(n) = A001157(n^2). - R. J. Mathar, Mar 31 2011
Dirichlet g.f. zeta(s)*zeta(s-2)*zeta(s-4)/zeta(2s-4). Dirichlet convolution of A001159 by the arithmetic function with terms n^2*A008966(n). - R. J. Mathar, Mar 31 2011
Sum_{k=1..n} a(k) ~ 189 * Zeta(3) * Zeta(5) * n^5 / Pi^6. - Vaclav Kotesovec, Feb 01 2019
Sum_{k>=1} 1/a(k) = 1.06464520174524878494847955427968776606386158167258511428260450690334042955... - Vaclav Kotesovec, Sep 20 2020

A211347 Numbers n such that n = sigma_k(m) for some k >= 1.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 24, 26, 28, 30, 31, 32, 33, 36, 38, 39, 40, 42, 44, 48, 50, 54, 56, 57, 60, 62, 63, 65, 68, 72, 73, 74, 78, 80, 82, 84, 85, 90, 91, 93, 96, 98, 102, 104, 108, 110, 112, 114, 120, 121, 122

Offset: 1

Jon Perry, Feb 05 2013

nonn

Sigma_k(n) = Sum[d|n, d^k].
Sigma_0(n) can be any positive integer and so is ignored in this sequence.
The asymptotic density of this sequence is 0 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020

			Sigma_2(4) = 1 + 4 + 16 = 21 so 21 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.
R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.
Eric W. Weisstein, MathWorld: Divisor function

Cf. A000203, A001157, A001158, A001159, A001160.
Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.
Cf. A000005.

upto[n_] := Select[Union@Flatten[{1, DivisorSigma[Range@Max[1,Floor@Log[#,n]], #] & /@ Range[2,n]}], # <= n &]; upto[122] (* Giovanni Resta, Feb 05 2013 *)

list(lim)=if(lim<3, return(if(lim<1,[],[1]))); my(v=List([1])); for(k=1,logint((lim\=1)-1,2), forfactored(m=2,sqrtnint(lim-1,k), my(t=sigma(m,k)); if(t<=lim, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Apr 09 2022

A279363 Sum of 4th powers of proper divisors of n.

Original entry on oeis.org

0, 1, 1, 17, 1, 98, 1, 273, 82, 642, 1, 1650, 1, 2418, 707, 4369, 1, 7955, 1, 10898, 2483, 14658, 1, 26482, 626, 28578, 6643, 41090, 1, 62644, 1, 69905, 14723, 83538, 3027, 133923, 1, 130338, 28643, 174994, 1, 236692, 1, 249170, 57893, 279858, 1, 423794, 2402, 401267, 83603, 485810, 1, 644372, 15267, 659842, 130403, 707298, 1, 1053636

Offset: 1

Ilya Gutkovskiy, Dec 10 2016

			a(10) = 1^4 + 2^4 + 5^4 = 642, because 10 has 3 proper divisors {1,2,5}.
a(11) = 1^4 = 1, because 11 has 1 proper divisor {1}.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Proper divisors.
Index entries for sequences related to sums of divisors

Cf. A000583, A001159.

Cf. A001065, A067558, A276634, A279364.

Table[DivisorSigma[4, n] - n^4, {n, 60}]

for(n=1, 60, print1(sigma(n, 4) - n^4,", ")) \\ Indranil Ghosh, Mar 18 2017

from sympy.ntheory import divisor_sigma
print([divisor_sigma(n,4) - n**4 for n in range(1,61)]) # Indranil Ghosh, Mar 18 2017

a(n) = 1 if n is prime.
a(p^k) = (p^(4*k) - 1)/(p^4 - 1) when p is prime.
Dirichlet g.f.: zeta(s-4)*(zeta(s) - 1).
a(n) = A001159(n) - A000583(n).
G.f.: -x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5 + Sum_{k>=1} k^4 x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 18 2017
Sum_{k=1..n} a(k) ~ (Zeta(5) - 1)*n^5 / 5. - Vaclav Kotesovec, Feb 02 2019

A363605 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^5.

Original entry on oeis.org

0, 1, 5, 16, 35, 76, 126, 226, 335, 531, 715, 1092, 1365, 1947, 2420, 3286, 3876, 5251, 5985, 7861, 8986, 11342, 12650, 16252, 17585, 21841, 24086, 29367, 31465, 38946, 40920, 49662, 53080, 62782, 66206, 80082, 82251, 97376, 102640, 120001, 123410, 146628

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A013663, A032741, A065608, A069153, A363604, A363606.

Cf. A000203, A001157, A001158, A001159.

a[n_] := DivisorSum[n, Binomial[# + 2, 4] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^5)))

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

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

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A372952 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} n/gcd(x_1, x_2, x_3, n).

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A082771 Triangular array, read by rows: t(n,k) = Sum_{d|n} d^k, 0 <= k < n.

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A279395 a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^4.

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A301548 Expansion of Product_{k>=1} (1 + x^k)^(sigma_4(k)).

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A386748 a(n) = n^3*sigma_4(n).

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A046871 Numbers k such that sigma_2(k) divides sigma_4(k).

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A065827 Sum of squares of divisors of square numbers.

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