A060724 -id:A060724 - cp's OEIS frontend

A054584 Number of subgroups of the group generated by a^n=1, b^3=1 and ab=ba.

2, 4, 6, 6, 4, 12, 4, 8, 10, 8, 4, 18, 4, 8, 12, 10, 4, 20, 4, 12, 12, 8, 4, 24, 6, 8, 14, 12, 4, 24, 4, 12, 12, 8, 8, 30, 4, 8, 12, 16, 4, 24, 4, 12, 20, 8, 4, 30, 6, 12, 12, 12, 4, 28, 8, 16, 12, 8, 4, 36, 4, 8, 20, 14, 8, 24, 4, 12, 12, 16, 4, 40, 4, 8, 18, 12, 8, 24, 4, 20, 18, 8, 4

Offset: 1

John W. Layman, Apr 12 2000

Also the number of subgroups of the group C_n X C_3 (where C_n is the cyclic group of order n). Number of subgroups of the group C_n X C_m is Sum_{i|n,j|m} gcd(i,j).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Hampejs, N. Holighaus, L. Tóth and C. Wiesmeyr, On the subgroups of the group Z_m X Z_n, arXiv preprint arXiv:1211.1797 [math.GR], 2012-2014. - From _N. J. A. Sloane_, Jan 02 2013

Cf. A060710, A060724, A062011, A060648, A035191, A000005.
Cf. A001620, A079978, A051176.
A row of A216624.

a054584 n = a000005 n + 3 * a079978 n * a000005 (a051176 n) + a035191 n
-- Reinhard Zumkeller, Aug 27 2012

for n from 1 to 500 do a := ifactors(n):s := 1:for k from 1 to nops(a[2]) do p := a[2][k][1]:e := a[2][k][2]: if p=3 then b := 2*e+1:else b := e+1:fi:s := s*b:od:printf(`%d,`,2*s); od:

f[d_ /; Mod[d, 3] == 0] = 4; f[] = 2; a[n] := Total[f /@ Divisors[n]]; Table[a[n], {n, 1, 100}](* Jean-François Alcover, Nov 21 2011, after Michael Somos *)
f[p_, e_] := e + 1; f[3, e_] := 2*e + 1; a[1] = 2; a[n_] := 2*Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 29 2022 *)

a(n)=if(n<1, 0, sumdiv(n,d, (d%3==0)*2+2)) /* Michael Somos, Sep 20 2005 */

a(n) = tau(n)+3*tau(n/3)+A035191(n) if n is congruent to 0 mod 3 else tau(n)+A035191(n), where A035191(n) is the number of divisors of n that are not congruent to 0 mod 3.
a(n)/2 is multiplicative with a(3^e)=2e+1 and a(p^e)=e+1 for p<>3.
Moebius transform is period 3 sequence [2, 2, 4, ...]. - Michael Somos, Sep 20 2005
G.f.: Sum_{k>0} x^k(2+2*x^k+4*x^(2k))/(1-x^(3k)).
From Amiram Eldar, Nov 29 2022: (Start)
Dirichlet g.f.: 2 * zeta(s)^2 * (1 + 1/3^s).
Sum_{k=1..n} a(k) ~ 2*(4*n*log(n) + (8*gamma - 4 - log(3))*n)/3, where gamma is Euler's constant (A001620). (End)

Additional comments from Vladeta Jovovic, Oct 25 2001

A064803 Number of subgroups of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

Original entry on oeis.org

1, 16, 28, 129, 64, 448, 116, 802, 445, 1024, 268, 3612, 368, 1856, 1792, 4387, 616, 7120, 764, 8256, 3248, 4288, 1108, 22456, 2607, 5888, 5776, 14964, 1744, 28672, 1988, 22308, 7504, 9856, 7424, 57405, 2816, 12224, 10304, 51328, 3448, 51968, 3788, 34572, 28480, 17728, 4516, 122836, 9009, 41712

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 21 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mario Hampejs and László Tóth, On the subgroups of finite Abelian groups of rank three, Annales Univ. Sci. Budapest., Sect. Comp. 39 (2013) 111-124.

Cf. A060648, A060724.

A064803 := proc(n)
    local a,f,nu,p,j ;
    a := 1 ;
    for f in ifactors(n)[2] do
        nu := op(2,f) ;
        p := op(1,f) ;
        add( (nu-floor((j-1)/2))*(2*j-floor((j-1)/2))*p^(2*nu-j),j=0..2*nu) ;
        a := a*% ;
    end do:
    a ;
end proc: # R. J. Mathar, May 11 2013

f[p_, e_] := Sum[(e - Floor[(j - 1)/2])*(2*j - Floor[(j - 1)/2])*p^(2*e - j), {j, 0, 2*e}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 29 2022 *)

For a prime p: a(p) = 2*(p^2+p+2). - Vladeta Jovovic, Oct 22 2001

Multiplicative with a(p^e) = Sum_{j=0..2*e} (e - floor((j - 1)/2))*(2*j - floor((j - 1)/2))*p^(2*e - j). - Amiram Eldar, Nov 29 2022

More terms from Laszlo Toth, May 11 2013

A344135 a(n) = Sum_{i|n, j|n, k|n} ijk/lcm(i,j,k).

Original entry on oeis.org

1, 14, 22, 93, 44, 308, 74, 472, 259, 616, 158, 2046, 212, 1036, 968, 2123, 344, 3626, 422, 4092, 1628, 2212, 602, 10384, 1227, 2968, 2548, 6882, 932, 13552, 1058, 9006, 3476, 4816, 3256, 24087, 1484, 5908, 4664, 20768, 1808, 22792, 1982, 14694, 11396, 8428, 2354, 46706, 3843, 17178, 7568

Offset: 1

Seiichi Manyama, May 10 2021

Cf. A060724, A344132, A344133, A344134.

a[n_]:= Sum[i*j*k/LCM[i,j,k], {i, (d = Divisors[n])}, {j, d}, {k, d}]; Array[a, 50] (* Amiram Eldar, May 10 2021 *)

a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, i*j*k/lcm([i, j, k]))));

If p is prime, a(p) = 4 + 3*p + p^2.

A280162 Number of subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 67, 212, 1983, 1120, 14204, 3652, 43339, 24033, 75040, 19156, 420396, 35872, 244684, 237440, 821335, 99472, 1610211, 152404, 2220960, 774224, 1283452, 318532, 9187868, 810969, 2403424, 2222704, 7241916, 783904, 15908480, 1016836, 14445411, 4061072, 6664624, 4090240, 47657439, 2031712

Offset: 1

Laszlo Toth, Dec 27 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems
G. A. Miller, On the subgroups of an abelian group, The Annals of Mathematics, 2nd Ser. 6:1 (1904), pp. 1-6.
L. Toth, The number of subgroups of the group Z_m x Z_n x Z_r x Z_s, arXiv:1611.03302 [math.GR], (2016).

Cf. A060724, A064803.

\\ For numsubgrp, see the Alekseyev link.
a(n)=my(f=factor(n)); prod(i=1,#f~, numsubgrp(f[i,1],f[i,2]*[1,1,1,1])) \\ Charles R Greathouse IV, Dec 27 2016

Terms a(32) and beyond from Charles R Greathouse IV, Dec 27 2016

A068984 a(n) = Sum_{d|n} d*tau(d)^2.

Original entry on oeis.org

1, 9, 13, 45, 21, 117, 29, 173, 94, 189, 45, 585, 53, 261, 273, 573, 69, 846, 77, 945, 377, 405, 93, 2249, 246, 477, 526, 1305, 117, 2457, 125, 1725, 585, 621, 609, 4230, 149, 693, 689, 3633, 165, 3393, 173, 2025, 1974, 837, 189, 7449, 470, 2214, 897

Offset: 1

Vladeta Jovovic, Apr 01 2002

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

Cf. A000005, A062369, A060724, A064950.

[&+[d*#Divisors(d)^2: d in Divisors(n)]:n in [1..51]]; // Marius A. Burtea, Sep 15 2019

a[n_] := DivisorSum[n, # * DivisorSigma[0, #]^2 &]; Array[a, 100] (* Amiram Eldar, Sep 15 2019 *)

a(n) = sumdiv(n, d, d*numdiv(d)^2) \\ Michel Marcus, Jun 17 2013

Multiplicative with a(p^e) = (p^(e+3)-3*p^(e+2)+4*p^(e+1)-p-1+2*p^(e+3)*e-6*p^(e+2)*e+4*p^(e+1)*e+p^(e+3)*e^2-2*p^(e+2)*e^2+p^(e+1)*e^2)/(p-1)^3.

A360430 Dirichlet convolution of Dedekind psi by A038040.

Original entry on oeis.org

1, 7, 10, 30, 16, 70, 22, 104, 63, 112, 34, 300, 40, 154, 160, 320, 52, 441, 58, 480, 220, 238, 70, 1040, 165, 280, 324, 660, 88, 1120, 94, 912, 340, 364, 352, 1890, 112, 406, 400, 1664, 124, 1540, 130, 1020, 1008, 490, 142, 3200, 315, 1155

Offset: 1

R. J. Mathar, Feb 07 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joël Bellaïche and Samit Dasgupta, The p-adic L-functions of evil Eisenstein series, Compositio Mathem. 151 (6) (2015), 999-1040, E_{k+2,psi,tau}.

Cf. A000005, A001615, A008966, A034718, A038040, A060724.

A360430 := proc(n)
    add(A001615(n/d)*numtheory[tau](d)*d,d=numtheory[divisors](n)) ;
end proc:

f[p_, e_] := ((e+2)*p + e)*(e+1)*p^(e-1)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 09 2023 *)

a(n) = Sum_{d|n} A001615(n/d)*A000005(d)*d.
Dirichlet g.f.: zeta(s-1)^3*zeta(s)/zeta(2*s).
Dirichlet convolution of A008966 by A034718.
Multiplicative with a(p^e) = ((e+2)*p + e)*(e+1)*p^(e-1)/2. - Amiram Eldar, Feb 09 2023

cp's OEIS Frontend

A054584 Number of subgroups of the group generated by a^n=1, b^3=1 and ab=ba.

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A064803 Number of subgroups of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

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A344135 a(n) = Sum_{i|n, j|n, k|n} i*j*k/lcm(i,j,k).

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A280162 Number of subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.

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A068984 a(n) = Sum_{d|n} d*tau(d)^2.

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A360430 Dirichlet convolution of Dedekind psi by A038040.

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A344135 a(n) = Sum_{i|n, j|n, k|n} ijk/lcm(i,j,k).