A001157 -id:A001157 - cp's OEIS frontend

A065741 Largest square <= sum of squares of divisors of n.

1, 4, 9, 16, 25, 49, 49, 81, 81, 121, 121, 196, 169, 225, 256, 324, 289, 441, 361, 529, 484, 576, 529, 841, 625, 841, 784, 1024, 841, 1296, 961, 1296, 1156, 1444, 1296, 1849, 1369, 1764, 1681, 2209, 1681, 2500, 1849, 2500, 2304, 2601, 2209, 3364, 2401

Offset: 1

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A001157, A048760, A065730-A065741.

Table[Floor[Sqrt[DivisorSigma[2, w]]//N]^2, {w, 1, 100}]

a(n) = { sqrtint(sigma(n,2))^2 } \\ Harry J. Smith, Oct 29 2009

a(n) = A048760(A001157(n)).

A067692 a(n) = Sum_{0 < d <= t <= n, d|n, t|n} d*t.

Original entry on oeis.org

1, 7, 13, 35, 31, 97, 57, 155, 130, 227, 133, 497, 183, 413, 418, 651, 307, 988, 381, 1155, 762, 953, 553, 2225, 806, 1307, 1210, 2093, 871, 3242, 993, 2667, 1762, 2183, 1802, 5096, 1407, 2705, 2418, 5155, 1723, 5858

Offset: 1

Reinhard Zumkeller, Feb 04 2002

Total area of all s X t rectangles, where the (s,t) are the pairs of divisors of n such that 1 <= s <= t. For example, when n = 4, the rectangles are 1 X 1, 1 X 2, 1 X 4, 2 X 2, 2 X 4, and 4 X 4, whose total area is a(4) = 1*1 + 1*2 + 1*4 + 2*2 + 2*4 + 4*4 = 35. - Wesley Ivan Hurt, Nov 15 2021

			a(6) = 1*1+1*2+1*3+1*6+2*2+2*3+2*6+3*3+3*6+6*6 = 1+2+3+6+4+6+12+9+18+36 = 97.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A000203, A001157, A002117, A060800.

Table[(DivisorSigma[1,n]^2+DivisorSigma[2,n])/2,{n,50}] (* Harvey P. Dale, Jan 31 2015 *)

a(n)=my(D=sigma(n));sumdiv(n,t,D-=t;t*(D+t)) \\ Charles R Greathouse IV, Aug 21 2011

a(n)=(sigma(n)^2+sigma(n,2))/2 \\ Charles R Greathouse IV, Aug 21 2011

a(n) = (1/2)*(sigma_1(n)^2 + sigma_2(n)), cf. A000203, A001157.
For p prime: a(p) = 1 + p + p^2, a(A000040(k)) = A060800(k).
Sum_{k=1..n} a(k) = (7/12)*zeta(3) * n^3 + O(n^2*log(n)^2). - Amiram Eldar, Dec 15 2023

A069153 a(n) = Sum_{d|n} d*(d-1)/2.

Original entry on oeis.org

0, 1, 3, 7, 10, 19, 21, 35, 39, 56, 55, 91, 78, 113, 118, 155, 136, 208, 171, 252, 234, 287, 253, 395, 310, 404, 390, 497, 406, 614, 465, 651, 586, 698, 626, 910, 666, 875, 822, 1060, 820, 1202, 903, 1239, 1144, 1289, 1081, 1643, 1197, 1581, 1414, 1736

Offset: 1

Benoit Cloitre, Apr 08 2002

Inverse Mobius transform of A000217. - R. J. Mathar, Jan 19 2009

			x^2 + 3*x^3 + 7*x^4 + 10*x^5 + 19*x^6 + 21*x^7 + 35*x^8 + 39*x^9 + 56*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A000203, A001157, A002117, A007437, A086666, A134840.

Cf. A032741, A065608, A363604, A363605, A363606.

with(numtheory):
seq((1/2)*(sigma[2](n) - sigma[1](n)), n = 1..100); # Peter Bala, Jan 21 2021

A069153[n_]:=Plus@@Binomial[Divisors[n],2];Array[A069153,100] (* Enrique Pérez Herrero, Feb 21 2012 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 - d) / 2)}

a(n) = my(f = factor(n)); (sigma(f, 2) - sigma(f)) / 2; \\ Amiram Eldar, Jan 01 2025

G.f.: Sum_{k>0} x^(2*k)/(1-x^k)^3. - Vladeta Jovovic, Dec 17 2002
Row sums of triangle A134840. - Gary W. Adamson, Nov 12 2007
G.f. A(x) = (1/2) * x * d/dx log( B(x) ) where B() is g.f. for A052847. - Michael Somos, Feb 12 2008
G.f.: Sum_{k>0} ((k^2 - k) / 2) * x^k / (1 - x^k). - Michael Somos, Feb 12 2008
From Peter Bala, Jan 21 2021: (Start)
a(n) = (1/2)*(sigma_2(n) - sigma_1(n)) = (1/2)*(A001157(n)  A000203(n)) =  (1/2)*A086666.
G.f.: A(x) = (1/2)* Sum_{n >= 1} x^(n^2)*( n*(n-1)*x^(3*n) - (n^2 + n - 2)*x^(2*n) + n*(3 - n)*x^n + n*(n - 1) )/(1 - x^n)^3. - differentiate equation 5 in Arndt twice w.r.t x and set x = 1. (End)
From Amiram Eldar, Jan 01 2025: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-2) - zeta(s-1)) / 2.
Sum_{k=1..n} a(k) ~ (zeta(3)/6) * n^3. (End)

A351307 Sum of the squares of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 17, 1, 1, 1, 17, 82, 1, 1, 17, 1, 1, 1, 273, 1, 82, 1, 17, 1, 1, 1, 17, 626, 1, 82, 17, 1, 1, 1, 273, 1, 1, 1, 1394, 1, 1, 1, 17, 1, 1, 1, 17, 82, 1, 1, 273, 2402, 626, 1, 17, 1, 82, 1, 17, 1, 1, 1, 17, 1, 1, 82, 4369, 1, 1, 1, 17, 1, 1, 1, 1394, 1, 1, 626, 17, 1, 1

Offset: 1

Wesley Ivan Hurt, Feb 06 2022

Inverse Möbius transform of n^2 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 20 2024

			a(16) = 273; a(16) = Sum_{d^2|16} (d^2)^2 = (1^2)^2 + (2^2)^2 + (4^2)^2 = 273.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), this sequence (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052, A247041 (zeta(5/2)), A008836, A001157.

f[p_, e_] := (p^(4*(1 + Floor[e/2])) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^4*x^k^2/(1-x^k^2))) \\ Seiichi Manyama, Feb 12 2022

from math import prod
from sympy import factorint
def A351307(n): return prod((p**(4+((e&-2)<<1))-1)//(p**4-1) for p,e in factorint(n).items()) # Chai Wah Wu, Jul 11 2024

a(n) = Sum_{d^2|n} (d^2)^2.
Multiplicative with a(p) = (p^(4*(1+floor(e/2))) - 1)/(p^4 - 1). - Amiram Eldar, Feb 07 2022
G.f.: Sum_{k>0} k^4*x^(k^2)/(1-x^(k^2)). - Seiichi Manyama, Feb 12 2022
From Amiram Eldar, Sep 19 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-4).
Sum_{k=1..n} a(k) ~ (zeta(5/2)/5) * n^(5/2). (End)
a(n) = Sum_{d|n} d^2 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 20 2024
a(n) = Sum_{d|n} lambda(d)*d^2*sigma_2(n/d), where lambda = A008836. - Ridouane Oudra, Jul 18 2025

A013963 a(n) = sigma_15(n), the sum of the 15th powers of the divisors of n.

Original entry on oeis.org

1, 32769, 14348908, 1073774593, 30517578126, 470199366252, 4747561509944, 35185445863425, 205891146443557, 1000030517610894, 4177248169415652, 15407492847694444, 51185893014090758, 155572843119354936, 437893920912786408, 1152956690052710401, 2862423051509815794

Offset: 1

N. J. A. Sloane

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 = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013674, A013954-A013972, A017665-A017712.

[DivisorSigma(15, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

DivisorSigma[15, Range[30]] (* Vincenzo Librandi, Sep 10 2016 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^15*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 15); \\ Amiram Eldar, Oct 29 2023

[sigma(n,15)for n in range(1,15)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^15*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-15)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(15*e+15)-1)/(p^15-1).
Sum_{k=1..n} a(k) = zeta(16) * n^16 / 16 + O(n^17). (End)

A013969 a(n) = sigma_21(n), the sum of the 21st powers of the divisors of n.

Original entry on oeis.org

1, 2097153, 10460353204, 4398048608257, 476837158203126, 21936961102828212, 558545864083284008, 9223376434903384065, 109418989141972712413, 1000000476837160300278, 7400249944258160101212, 46005141850728850805428, 247064529073450392704414, 1171356134499851307229224

Offset: 1

N. J. A. Sloane

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 = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.

[DivisorSigma(21,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

Table[DivisorSigma[21,n],{n,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

vector(50, n, sigma(n,21)) \\ G. C. Greubel, Nov 03 2018

[sigma(n,21)for n in range(1,13)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^21*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Sum_{n>=1} a(n)/exp(2*Pi*n) = 77683/552 = Bernoulli(22)/44. - Vaclav Kotesovec, May 07 2023
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(21*e+21)-1)/(p^21-1).
Dirichlet g.f.: zeta(s)*zeta(s-21).
Sum_{k=1..n} a(k) = zeta(22) * n^22 / 22 + O(n^23). (End)

A066135 a(n) = least number m > 1 such that sigma_n(m) = k*m for some k.

Original entry on oeis.org

6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 194, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 228, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 194, 6, 10, 6

Offset: 1

Labos Elemer, Dec 06 2001

nonn

a(n) <= 2p, where p = A002586(n) is the smallest prime factor of (1 + 2^n). (Proof. Since sigma_n(2p) = (1 + 2^n)(1 + p^n) and p is odd, 2p divides sigma_n(2p).) - Jonathan Sondow, Nov 23 2012

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A001157, A001158, A001159, A002586, A007691, A046762-A046764, A055709-A055717.

Cf. A218860, A218861 (unique values and where they first occur).

Table[m = 2; While[Mod[DivisorSigma[n, m], m] > 0, m++]; m, {n, 100}] (* T. D. Noe, Nov 23 2012 *)

Sum{d^n} = ka(n), d runs over the divisors of a(n), where k is an integer and a(n) is the smallest suitable number.

Definition and formulas corrected by Jonathan Sondow, Nov 23 2012

A328259 a(n) = n * sigma_2(n).

Original entry on oeis.org

1, 10, 30, 84, 130, 300, 350, 680, 819, 1300, 1342, 2520, 2210, 3500, 3900, 5456, 4930, 8190, 6878, 10920, 10500, 13420, 12190, 20400, 16275, 22100, 22140, 29400, 24418, 39000, 29822, 43680, 40260, 49300, 45500, 68796, 50690, 68780, 66300, 88400, 68962, 105000, 79550, 112728, 106470

Offset: 1

Ilya Gutkovskiy, Oct 09 2019

Moebius transform of A027847.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A001157, A027847, A038040, A064987, A275585, A281372, A294362.

Table[n DivisorSigma[2, n], {n, 1, 45}]
nmax = 45; CoefficientList[Series[Sum[k^3 x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = n*sigma(n, 2); \\ Michel Marcus, Dec 02 2020

G.f.: Sum_{k>=1} k^3 * x^k / (1 - x^k)^2.
G.f.: Sum_{k>=1} k * x^k * (1 + 4 * x^k + x^(2*k)) / (1 - x^k)^4.
Dirichlet g.f.: zeta(s - 1) * zeta(s - 3).
Sum_{k=1..n} a(k) ~ zeta(3) * n^4 / 4. - Vaclav Kotesovec, Oct 09 2019
Multiplicative with a(p^e) = (p^(3*e+2) - p^e)/(p^2 - 1). - Amiram Eldar, Dec 02 2020
G.f.: Sum_{n >= 1} q^(n^2)*( n^4 - (2*n^4 - 4*n^3 - 3*n^2 - n)*q^n - (8*n^3 - 4*n)*q^(2*n) + (2*n^4 + 4*n^3 - 3*n^2 + n)*q^(3*n) - n^4*q^(4*n) )/(1 - q^n)^4. Apply the operator x*d/dx twice, followed by the operator q*d/dq once, to equation 5 in Arndt and then set x = 1. - Peter Bala, Jan 21 2021
a(n) = Sum_{k = 1..n} sigma_3( gcd(k, n) ) = Sum_{d divides n} sigma_3(d) * phi(n/d). - Peter Bala, Jan 19 2024
a(n) = Sum_{1 <= i, j, k <= n} sigma_1( gcd(i, j, k, n) ) = Sum_{d divides n} sigma_1(d) * J_3(n/d), where the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 22 2024

A007433 Inverse Moebius transform applied twice to squares.

Original entry on oeis.org

1, 6, 11, 27, 27, 66, 51, 112, 102, 162, 123, 297, 171, 306, 297, 453, 291, 612, 363, 729, 561, 738, 531, 1232, 678, 1026, 922, 1377, 843, 1782, 963, 1818, 1353, 1746, 1377, 2754, 1371, 2178, 1881, 3024, 1683

Offset: 1

N. J. A. Sloane

Dirichlet convolution of A001157 and A000012. Dirichlet convolution of A000005 and A000290 (Jovovic formula). - R. J. Mathar, Feb 03 2011

Sum of the squares of the divisors d1 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022

			G.f. = x + 6*x^2 + 11*x^3 + 27*x^4 + 27*x^5 + 66*x^6 + 51*x^7 + 112*x^8 + 102*x^9 + ... - _Michael Somos_, Jul 15 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A134577.

a[n_] := Plus @@ DivisorSigma[2, Divisors[n]]; Array[a, 41] (* Robert G. Wilson v, May 05 2010 *)
a[ n_] := If[ n < 1, 0, Times @@ (If[ # == 1, 1, (#^(2 #2 + 4) - (#2 + 2) #^2 + #2 + 1) / (#^2 - 1)^2] & @@@ FactorInteger @ n)]; (* Michael Somos, Jul 15 2018 *)

/* Dirichlet convolution of A001157, A000012 (Mathar): */
a(n)=sumdiv(n, d, sigma(d,2))

/* Dirichlet convolution of A000005, A000290 (Mathar): */
a(n)=sumdiv(n, d, d^2*sigma(n/d,0))

a(n) = Sum_{d|n} d^2*tau(n/d). - Vladeta Jovovic, Jul 31 2002
Equals A134577 * [1, 2, 3, ...]. - Gary W. Adamson, Nov 02 2007
G.f.: Sum_{k>=1} sigma_2(k)*x^k/(1 - x^k), where sigma_2(k) is the sum of squares of divisors of k (A001157). - Ilya Gutkovskiy, Jan 16 2017
Dirichlet g.f.: zeta(s-2)*zeta(s)^2. - Benedict W. J. Irwin, Jul 14 2018
a(n) is multiplicative with a(p^e) = (p^(2*e + 4) - (e+2) * p^2 + e+1) / (p^2 - 1)^2. - Michael Somos, Jul 15 2018
Sum_{k=1..n} a(k) ~ Zeta(3)^2 * n^3 / 3. - Vaclav Kotesovec, Nov 04 2018

a(38) corrected by Ilya Gutkovskiy, Jan 16 2016

A013960 a(n) = sigma_12(n), the sum of the 12th powers of the divisors of n.

Original entry on oeis.org

1, 4097, 531442, 16781313, 244140626, 2177317874, 13841287202, 68736258049, 282430067923, 1000244144722, 3138428376722, 8918294543346, 23298085122482, 56707753666594, 129746582562692, 281543712968705, 582622237229762, 1157115988280531, 2213314919066162

Offset: 1

N. J. A. Sloane

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 = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013671, A013954-A013972, A017665-A017712.

[DivisorSigma(12, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

DivisorSigma[12,Range[20]] (* Harvey P. Dale, Jan 28 2015 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^12*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 12); \\ Amiram Eldar, Oct 29 2023

[sigma(n,12) for n in range(1,17)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^12*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-12)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(12*e+12)-1)/(p^12-1).
Sum_{k=1..n} a(k) = zeta(13) * n^13 / 13 + O(n^14). (End)

cp's OEIS Frontend

A065741 Largest square <= sum of squares of divisors of n.

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A067692 a(n) = Sum_{0 < d <= t <= n, d|n, t|n} d*t.

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A069153 a(n) = Sum_{d|n} d*(d-1)/2.

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A351307 Sum of the squares of the square divisors of n.

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A013963 a(n) = sigma_15(n), the sum of the 15th powers of the divisors of n.

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A013969 a(n) = sigma_21(n), the sum of the 21st powers of the divisors of n.

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A066135 a(n) = least number m > 1 such that sigma_n(m) = k*m for some k.

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