A062952 -id:A062952 - cp's OEIS frontend

A038040 a(n) = n*d(n), where d(n) = number of divisors of n (A000005).

1, 4, 6, 12, 10, 24, 14, 32, 27, 40, 22, 72, 26, 56, 60, 80, 34, 108, 38, 120, 84, 88, 46, 192, 75, 104, 108, 168, 58, 240, 62, 192, 132, 136, 140, 324, 74, 152, 156, 320, 82, 336, 86, 264, 270, 184, 94, 480, 147, 300, 204, 312, 106, 432, 220, 448, 228, 232, 118

Offset: 1

Christian G. Bower

Dirichlet convolution of sigma(n) (A000203) with phi(n) (A000010). - Michael Somos, Jun 08 2000
Dirichlet convolution of f(n)=n with itself. See the Apostol reference for Dirichlet convolutions. - Wolfdieter Lang, Sep 09 2008
Sum of all parts of all partitions of n into equal parts. - Omar E. Pol, Jan 18 2013

			For n = 6 the partitions of 6 into equal parts are [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. The sum of all parts is 6 + 3 + 3 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 = 24 equalling 6 times the number of divisors of 6, so a(6) = 24. - _Omar E. Pol_, May 08 2021

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 29 ff.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 162.

T. D. Noe, Table of n, a(n) for n = 1..1000
J. Bourgain, S. V. Konyagin and I. E. Shparlinski, Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithms, Int. Math. Res. Notices, 2008 (2008), Art. ID rnn 090, 1-29.
Jean Bourgain, Sergei Konyagin and Igor Shparlinski. Distribution on elements of cosets of small subgroups and applications, arXiv:1103.0567 [math.NT], Mar 2 2011.
Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 147. [Broken link?]
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 147.

Cf. A000005, A000010, A000203, A001620, A018804, A029935, A064987, A062952.
Cf. A127648, A127093, A127528.
Cf. A038044, A143127 (partial sums), A328722 (Dirichlet inverse).
Column 1 of A329323.

a038040 n = a000005 n * n  -- Reinhard Zumkeller, Jan 21 2014

with(numtheory): A038040 := n->tau(n)*n;

a[n_] := DivisorSigma[0, n]*n; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Sep 03 2012 *)

n*numlib::tau (n)$ n=1..90 // Zerinvary Lajos, May 13 2008

a(n)=if(n<1,0,direuler(p=2,n,1/(1-p*X)^2)[n])

a(n)=if(n<1,0,polcoeff(sum(k=1,n,k*x^k/(x^k-1)^2,x*O(x^n)),n)) /* Michael Somos, Jan 29 2005 */

a(n) = n*numdiv(n); \\ Michel Marcus, Oct 24 2020

from sympy import divisor_count as d
def a(n): return n*d(n)
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Mar 15 2022

[n*sigma(n,0) for n in range(1, 60)] # Stefano Spezia, Jul 20 2025

Dirichlet g.f.: zeta(s-1)^2.
G.f.: Sum_{n>=1} n*x^n/(1-x^n)^2. - Vladeta Jovovic, Dec 30 2001
Sum_{k=1..n} sigma(gcd(n, k)). Multiplicative with a(p^e) = (e+1)*p^e. - Vladeta Jovovic, Oct 30 2001
Equals A127648 * A127093 * the harmonic series, [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, May 10 2007
Equals row sums of triangle A127528. - Gary W. Adamson, May 21 2007
a(n) = n*A000005(n) = A066186(n) - n*(A000041(n) - A000005(n)) = A066186(n) - n*A144300(n). - Omar E. Pol, Jan 18 2013
a(n) = A000203(n) * A240471(n) + A106315(n). - Reinhard Zumkeller, Apr 06 2014
L.g.f.: Sum_{k>=1} x^k/(1 - x^k) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 13 2017
a(n) = Sum_{d|n} A018804(d). - Amiram Eldar, Jun 23 2020
a(n) = Sum_{d|n} phi(d)*sigma(n/d). - Ridouane Oudra, Jan 21 2021
G.f.: Sum_{n >= 1} q^(n^2)*(n^2 + 2*n*q^n - n^2*q^(2*n))/(1 - q^n)^2. - Peter Bala, Jan 22 2021
a(n) = Sum_{k=1..n} sigma(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ x/sqrt(log x). That is, there are 0 < A < B such that Ax/sqrt(log x) < f(x) < Bx/sqrt(log x). - Charles R Greathouse IV, Mar 15 2022
Sum_{k=1..n} a(k) ~ n^2*log(n)/2 + (gamma - 1/4)*n^2, where gamma is Euler's constant (A001620). - Amiram Eldar, Oct 25 2022
Mobius transform of A060640. - R. J. Mathar, Feb 07 2023

A062354 a(n) = sigma(n)*phi(n).

Original entry on oeis.org

1, 3, 8, 14, 24, 24, 48, 60, 78, 72, 120, 112, 168, 144, 192, 248, 288, 234, 360, 336, 384, 360, 528, 480, 620, 504, 720, 672, 840, 576, 960, 1008, 960, 864, 1152, 1092, 1368, 1080, 1344, 1440, 1680, 1152, 1848, 1680, 1872, 1584, 2208, 1984, 2394, 1860

Offset: 1

Jason Earls, Jul 06 2001

Let G_n be the group of invertible 2 X 2 matrices mod n (sequence A000252). a(n) is the number of conjugacy classes in G_n. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 13 2001
a(n) = Sum_{d|n} phi(n*d). - Vladeta Jovovic, Apr 17 2002
Apparently the Mobius transform of A062952. - R. J. Mathar, Oct 01 2011

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, Prob. 7.2 12, p. 141.

T. D. Noe, Table of n, a(n) for n=1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (Pi^2 * n^3 / 18) for n = 1..1000000
J.-L. Nicolas and Jonathan Sondow, Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis, arXiv:1211.6944 [math.HO], 2012, to appear in RAMA125 Proceedings, Contemp. Math.

Cf. A000010, A000203, A000252, A062355, A064840, A093827.

Table[EulerPhi[n] DivisorSigma[1, n], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

a(n)=sigma(n)*eulerphi(n); vector(150,n,a(n))

Multiplicative with a(p^e) = p^(e-1)*(p^(e+1)-1). - Vladeta Jovovic, Apr 17 2002
Dirichlet g.f.: zeta(s-1)*zeta(s-2)*product_{primes p} (1-p^(1-s)-p^(-s)+p^(2-2s)). - R. J. Mathar, Oct 01 2011, corrected by Vaclav Kotesovec, Dec 17 2019
6/Pi^2 < a(n)/n^2 < 1 for n > 1. - Jonathan Sondow, Mar 06 2014
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^3 / 18, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.535896... - Vaclav Kotesovec, Dec 17 2019
Sum_{n>=1} 1/a(n) = 1.7865764... (A093827). - Amiram Eldar, Aug 20 2020
a(n)/n^2 > 8/Pi^2 for odd n. - M. F. Hasler, Jul 08 2025

A127466 Triangle read by rows: A054525 * A127481 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 3, 0, 6, 4, 4, 0, 8, 5, 0, 0, 0, 20, 6, 6, 12, 0, 0, 12, 7, 0, 0, 0, 0, 0, 42, 8, 8, 0, 16, 0, 0, 0, 32, 9, 0, 18, 0, 0, 0, 0, 0, 54, 10, 10, 0, 0, 40, 0, 0, 0, 0, 40

Offset: 1

Gary W. Adamson, Jan 15 2007

Mobius transform of A127481.

			First few rows of the triangle are:
1;
2, 2;
3, 0, 6;
4, 4, 0, 8;
5, 0, 0, 0, 20;
6, 6, 12, 0, 0, 12;
7, 0, 0, 0, 0, 0, 42;
8, 8, 0, 16, 0, 0, 0, 32;
...

Cf. A054522, A127093, A062952, A002618.

Cf. A127093, A127481, A054525.

A127466 := proc(n,k)
    add( A054525(n,j)*A127481(j,k),j=k..n) ;
end proc: # R. J. Mathar, Sep 06 2013

Sum_{k=1..n} T(n,k) = n^2.

T(n,n) = A002618(n) = n*phi(n).

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Aug 23 2007

A373133 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^3 ).

Original entry on oeis.org

1, 106, 1041, 7218, 19345, 110346, 136801, 465522, 768327, 2050570, 1947121, 7513938, 5226481, 14500906, 20138145, 29822066, 25640641, 81442662, 49651921, 139632210, 142409841, 206394826, 154751521, 484608402, 302749845, 554006986, 560366223, 987429618, 616040881

Offset: 1

Seiichi Manyama, May 26 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A059376, A084220.
Cf. A373130, A373131.
Cf. A062952, A373132, A373135.
Cf. A013662, A013665.

f[p_, e_] := (p^(6*e+4)*(p+1) - p^(3*e)*(p^4+p^3+p+1) + p^2+p)/((p^2-1)*(p^3+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=3, m=3) = sumdiv(n, d, J(d, k)*sigma(d^m));

a(n) = Sum_{d|n} J_3(d) * sigma(d^3), where the Jordan totient function J_3(n) = A059376(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(6*e+4)*(p+1) - p^(3*e)*(p^4+p^3+p+1) + p^2+p)/((p^2-1)*(p^3+1)).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(4) * zeta(7) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 1/p^4 - 1/p^5 - 1/p^6 - 1/p^7 + 1/p^8) = 1.71945569563704656468... . (End)

A372227 a(n) = Sum_{k=1..n} sigma( (n/gcd(k,n))^2 ).

Original entry on oeis.org

1, 8, 27, 70, 125, 216, 343, 578, 753, 1000, 1331, 1890, 2197, 2744, 3375, 4666, 4913, 6024, 6859, 8750, 9261, 10648, 12167, 15606, 15745, 17576, 20427, 24010, 24389, 27000, 29791, 37418, 35937, 39304, 42875, 52710, 50653, 54872, 59319, 72250, 68921, 74088

Offset: 1

Seiichi Manyama, May 19 2024

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

Cf. A062952, A372998, A373000.
Cf. A062380, A372996.
Cf. A005117.
Cf. A002117, A182448.

a[n_] := DivisorSum[n, EulerPhi[#] * DivisorSigma[1, #^2] &]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d^2));

If k is squarefree (cf. A005117) then a(k) = k^3.
a(n) = Sum_{d|n} phi(d) * sigma(d^2).
From Amiram Eldar, May 20 2024: (Start)
Multiplicative with a(p^e) = (p^(3*e+3)-1)/(p^3-1) - (p^e-1)/(p-1).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = (Pi^2/15) * zeta(3) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.03291869994469216597... . (End)

A373132 a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( ( n/gcd(x_1, x_2, n) )^2 ).

Original entry on oeis.org

1, 22, 105, 394, 745, 2310, 2737, 6490, 8817, 16390, 15961, 41370, 30745, 60214, 78225, 104602, 88417, 193974, 137161, 293530, 287385, 351142, 291985, 681450, 469345, 676390, 717081, 1078378, 731641, 1720950, 953281, 1676698, 1675905, 1945174, 2039065, 3473898

Offset: 1

Seiichi Manyama, May 26 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007434, A084218.
Cf. A062952, A373133, A373135.
Cf. A002117, A013663.

f[p_, e_] := (p^(4*e+3)*(p+1) - p^(2*e)*(p^3+p^2+p+1) + p^2 + p)/(p^4-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=2, m=2) = sumdiv(n, d, J(d, k)*sigma(d^m));

a(n) = Sum_{d|n} J_2(d) * sigma(d^2), where the Jordan totient function J_2(n) = A007434(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(4*e+3)*(p+1) - p^(2*e)*(p^3+p^2+p+1) + p^2 + p)/(p^4-1).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(3) * zeta(5) * Product_{p prime} (1 + 1/p^2 - 1/p^3 - 1/p^4 - 1/p^5 + 1/p^6) = 1.489916841715667671605... . (End)

A373135 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} sigma( ( n/gcd(x_1, x_2, x_3, x_4, n) )^4 ).

Original entry on oeis.org

1, 466, 9681, 123106, 487345, 4511346, 6722401, 31576546, 63779361, 227102770, 235777201, 1191789186, 883674961, 3132638866, 4717986945, 8084578786, 7411648321, 29721182226, 17926949521, 59995093570, 65079564081, 109872175666, 81870270241, 305692541826

Offset: 1

Seiichi Manyama, May 26 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A059377, A372966.
Cf. A062952, A373132, A373133.
Cf. A013663, A013667.

f[p_, e_] := (p^(8*e+5)*(p+1) - p^(4*e)*(p^5+p^4+p+1) + p^2 + p)/((p^2-1)*(p^4+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=4, m=4) = sumdiv(n, d, J(d, k)*sigma(d^m));

a(n) = Sum_{d|n} J_4(d) * sigma(d^4), where the Jordan totient function J_4(n) = A059377(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(8*e+5)*(p+1) - p^(4*e)*(p^5+p^4+p+1) + p^2 + p)/((p^2-1)*(p^4+1)).
Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = zeta(5) * zeta(9) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 - 1/p^5 - 1/p^6 - 1/p^7 - 1/p^8 - 1/p^9 + 1/p^10) = 1.83382546873826519758... . (End)

A341637 a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

Original entry on oeis.org

1, 6, 12, 30, 30, 72, 56, 138, 123, 180, 132, 360, 182, 336, 360, 602, 306, 738, 380, 900, 672, 792, 552, 1656, 795, 1092, 1176, 1680, 870, 2160, 992, 2538, 1584, 1836, 1680, 3690, 1406, 2280, 2184, 4140, 1722, 4032, 1892, 3960, 3690, 3312, 2256, 7224, 2835, 4770, 3672, 5460

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

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

Cf. A000010, A000203, A000385, A034761, A038040, A062354, A062952, A183699, A330523.

Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[1, n/d], {d, Divisors[n]}], {n, 52}]
Table[Sum[DivisorSigma[1, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 52}]
f[p_, e_] := (p^(2*e + 3) - (e + 1)*(p^2 - 1)*p^e - p)/((p - 1)^2*(p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 12 2022 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*sigma(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} sigma(gcd(n,k)) * sigma(n/gcd(n,k)).
From Amiram Eldar, Nov 12 2022: (Start)
Multiplicative with a(p^e) = (p^(2*e+3) - (e+1)*(p^2-1)*p^e - p)/((p-1)^2*(p+1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)/3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/3) * A183699 * A330523 = 0.581007... . (End)

A341638 a(n) = Sum_{d|n} phi(d) * sigma(d) * tau(n/d).

Original entry on oeis.org

1, 5, 10, 23, 26, 50, 50, 101, 97, 130, 122, 230, 170, 250, 260, 427, 290, 485, 362, 598, 500, 610, 530, 1010, 671, 850, 904, 1150, 842, 1300, 962, 1761, 1220, 1450, 1300, 2231, 1370, 1810, 1700, 2626, 1682, 2500, 1850, 2806, 2522, 2650, 2210, 4270, 2493, 3355, 2900, 3910, 2810, 4520

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

Inverse Moebius transform of A062952.

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

Cf. A000005, A000010, A000203, A007430, A062354, A062952, A191831.

Cf. A002117, A013661, A330523.

Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 54}]
Table[Sum[DivisorSigma[0, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 54}]
f[p_, e_] := (p^(2*e + 4) - p^(e + 3) - 2*p^(e + 2) - p^(e + 1) + (e + 1)*p^3 - (e - 1)*p + 1)/(p^2 - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 26 2023 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*numdiv(n/d)); \\ Michel Marcus, Feb 17 2021

a(n) = Sum_{k=1..n} tau(gcd(n,k)) * sigma(n/gcd(n,k)).
a(n) = Sum_{d|n} A062952(d).
a(n) = Sum_{k=1..n} tau(n/gcd(n,k))*sigma(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 09 2021
From Amiram Eldar, Jan 26 2023: (Start)
Multiplicative with a(p^e) = (p^(2*e+4) - p^(e+3) - 2*p^(e+2) - p^(e+1) + (e+1)*p^3 - (e-1)*p + 1)/(p^2-1)^2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)^2/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A013661 * A002117^2 * A330523 / 3 = 0.424578... . (End)

A372226 a(n) = Sum_{k=1..n} sigma_2( n/gcd(k,n) ).

Original entry on oeis.org

1, 6, 21, 48, 105, 126, 301, 388, 567, 630, 1221, 1008, 2041, 1806, 2205, 3116, 4641, 3402, 6517, 5040, 6321, 7326, 11661, 8148, 13125, 12246, 15327, 14448, 23577, 13230, 28861, 24956, 25641, 27846, 31605, 27216, 49321, 39102, 42861, 40740, 67281, 37926

Offset: 1

Seiichi Manyama, May 19 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A062949, A062952.
Cf. A064987.
Cf. A002117, A013662.

a[n_] := DivisorSum[n, EulerPhi[#] * DivisorSigma[2, #] &]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d, 2));

a(n) = Sum_{d|n} phi(d) * sigma_2(d).
From Amiram Eldar, May 20 2024: (Start)
Multiplicative with a(p^e) = (p^(3*e+5) - p^(3*e+4) - p^(e+3) + p^e + p^4 - p^2) / ((p^2 - 1) * (p^3 - 1)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.749582840863254826301... . (End)

cp's OEIS Frontend

A038040 a(n) = n*d(n), where d(n) = number of divisors of n (A000005).

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A062354 a(n) = sigma(n)*phi(n).

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A127466 Triangle read by rows: A054525 * A127481 as infinite lower triangular matrices.

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A373133 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^3 ).

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A372227 a(n) = Sum_{k=1..n} sigma( (n/gcd(k,n))^2 ).

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A373132 a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( ( n/gcd(x_1, x_2, n) )^2 ).

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A373135 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} sigma( ( n/gcd(x_1, x_2, x_3, x_4, n) )^4 ).

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