A060648 -id:A060648 - cp's OEIS frontend

A203444 Numbers in range of Dedekind Psi function: A001615.

1, 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 36, 38, 42, 44, 48, 54, 56, 60, 62, 68, 72, 74, 80, 84, 90, 96, 98, 102, 104, 108, 110, 112, 114, 120, 126, 128, 132, 138, 140, 144, 150, 152, 158, 160, 162, 164, 168, 174, 176, 180, 182, 186, 192, 194, 198, 200

Offset: 1

Enrique Pérez Herrero, Jan 02 2012

nonn

a(n) is even for n>2

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Patrick Sole and Michel Planat, Extreme values of the Dedekind Psi function, J. Comb. Number Theory 3 (2011), no. 1, 33-38, arXiv:1011.1825v2.

Cf. A001615, A175836, A060648.
Cf. A007617.
Cf. A002202.

terms = 100; Clear[seq]; seq[k_] := seq[k] = Table[DirichletConvolve[j, MoebiusMu[j]^2, j, n], {n, 1, k terms}] // Union // PadRight[#, terms]&;
seq[k = 1]; seq[k++]; While[Print[k]; seq[k] != seq[k-1], k++];
seq[k] (* Jean-François Alcover, Dec 14 2018, after Jan Mangaldan in A001615 *)

A344222 a(n) = Sum_{k=1..n} tau(gcd(k,n)^4), where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 6, 7, 16, 9, 42, 11, 36, 25, 54, 15, 112, 17, 66, 63, 76, 21, 150, 23, 144, 77, 90, 27, 252, 49, 102, 79, 176, 33, 378, 35, 156, 105, 126, 99, 400, 41, 138, 119, 324, 45, 462, 47, 240, 225, 162, 51, 532, 81, 294, 147, 272, 57, 474, 135, 396, 161, 198, 63, 1008, 65, 210, 275, 316, 153, 630, 71

Offset: 1

Seiichi Manyama, May 12 2021

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

Cf. A060648, A072691, A344221, A344223, A344224, A344225.

Table[Sum[DivisorSigma[0,GCD[k,n]^4],{k,n}],{n,100}] (* Giorgos Kalogeropoulos, May 13 2021 *)
f[p_, e_] := (p^e*(p + 3) - 4)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sum(k=1, n, numdiv(gcd(k, n)^4));

a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^4));

PARI
```
a(n) = n*sumdiv(n, d, 4^omega(d)/d);
```

a(n) = Sum_{d|n} phi(n/d) * tau(d^4).
a(n) = n * Sum_{d|n} 4^omega(d) / d.
If p is prime, a(p) = 4 + p.
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = (p^e*(p + 3) - 4)/(p - 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 3/p^2) = 2.4997873122... . (End)

A344304 Number of cyclic subgroups of the group (C_n)^8, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 256, 3281, 32896, 97657, 839936, 960801, 4210816, 7176641, 25000192, 21435889, 107931776, 67977561, 245965056, 320412617, 538984576, 435984841, 1837220096, 943531281, 3212524672, 3152388081, 5487587584, 3559590241, 13815687296, 7629472657, 17402255616

Offset: 1

Seiichi Manyama, May 14 2021

Inverse Moebius transform of A160908.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Cf. A000010, A013666, A060648, A064969, A160908, A280184, A344219, A344302, A344303, A344305, A344306.

f[p_, e_] := 1 + ((p^8 - 1)/(p - 1))*((p^(7*e) - 1)/(p^7 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)

a160908(n) = sumdiv(n, d, moebius(n/d)*d^8)/eulerphi(n);
a(n) = sumdiv(n, d, a160908(d));

a(n) = Sum_{x_1|n, x_2|n, ..., x_8|n} phi(x_1)*phi(x_2)* ... *phi(x_8)/phi(lcm(x_1, x_2, ..., x_8)).
If p is prime, a(p) = 1 + (p^8 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^8 - 1)/(p - 1))*((p^(7*e) - 1)/(p^7 - 1)).
Sum_{k=1..n} a(k) ~ c * n^8, where c = (zeta(8)/8) * Product_{p prime} ((1-1/p^7)/(p^2*(1-1/p))) = 0.2432888374... . (End)

A344305 Number of cyclic subgroups of the group (C_n)^9, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 512, 9842, 131328, 488282, 5039104, 6725602, 33620224, 64576643, 250000384, 235794770, 1292530176, 883708282, 3443508224, 4805671444, 8606777600, 7411742282, 33063241216, 17927094322, 64125098496, 66193374884, 120726922240, 81870575522, 330890244608

Offset: 1

Seiichi Manyama, May 14 2021

Inverse Moebius transform of A160953.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Cf. A000010, A013667, A160953, A060648, A064969, A280184, A344219, A344302, A344303, A344304, A344306.

f[p_, e_] := 1 + ((p^9 - 1)/(p - 1))*((p^(8*e) - 1)/(p^8 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)

a160953(n) = sumdiv(n, d, moebius(n/d)*d^9)/eulerphi(n);
a(n) = sumdiv(n, d, a160953(d));

a(n) = Sum_{x_1|n, x_2|n, ..., x_9|n} phi(x_1)*phi(x_2)* ... *phi(x_9)/phi(lcm(x_1, x_2, ..., x_9)).
If p is prime, a(p) = 1 + (p^9 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^9 - 1)/(p - 1))*((p^(8*e) - 1)/(p^8 - 1)).
Sum_{k=1..n} a(k) ~ c * n^9, where c = (zeta(9)/9) * Product_{p prime} ((1-1/p^8)/(p^2*(1-1/p))) = 0.2161023934... . (End)

A344306 Number of cyclic subgroups of the group (C_n)^10, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 1024, 29525, 524800, 2441407, 30233600, 47079209, 268698112, 581150417, 2500000768, 2593742461, 15494720000, 11488207655, 48209110016, 72082541675, 137573433856, 125999618779, 595098027008, 340614792101, 1281250393600

Offset: 1

Seiichi Manyama, May 14 2021

Inverse Moebius transform of A160957.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Cf. A000010, A013668, A160957, A060648, A064969, A280184, A344219, A344302, A344303, A344304, A344305.

f[p_, e_] := 1 + ((p^10 - 1)/(p - 1))*((p^(9*e) - 1)/(p^9 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)

a160957(n) = sumdiv(n, d, moebius(n/d)*d^10)/eulerphi(n);
a(n) = sumdiv(n, d, a160957(d));

a(n) = Sum_{x_1|n, x_2|n, ..., x_10|n} phi(x_1)*phi(x_2)* ... *phi(x_10)/phi(lcm(x_1, x_2, ..., x_10)).
If p is prime, a(p) = 1 + (p^10 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^10 - 1)/(p - 1))*((p^(9*e) - 1)/(p^9 - 1)).
Sum_{k=1..n} a(k) ~ c * n^10, where c = (zeta(10)/10) * Product_{p prime} ((1-1/p^9)/(p^2*(1-1/p))) = 0.1944248708... . (End)

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

Original entry on oeis.org

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

A062368 Multiplicative with a(p^e) = (e+1)(e+2)(4*e+3)/6.

Original entry on oeis.org

1, 7, 7, 22, 7, 49, 7, 50, 22, 49, 7, 154, 7, 49, 49, 95, 7, 154, 7, 154, 49, 49, 7, 350, 22, 49, 50, 154, 7, 343, 7, 161, 49, 49, 49, 484, 7, 49, 49, 350, 7, 343, 7, 154, 154, 49, 7, 665, 22, 154, 49, 154, 7, 350, 49, 350, 49, 49, 7, 1078, 7, 49, 154, 252, 49, 343, 7, 154

Offset: 1

Vladeta Jovovic, Jul 07 2001

Conjecture: this is the third inverse Mobius transform of the sequence 4^A001221(n). - R. J. Mathar, Aug 09 2012

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A001221, A002412, A007425, A060648, A062380.

f[p_, e_] := (e+1)*(e+2)*(4*e+3)/6; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 05 2023 *)

a(n) = Sum_{i|n, j|n} tau(i)*tau(j)/tau(gcd(i, j)), where tau(n) = number of divisors of n, cf. A000005.
Also a(n) = Sum_{i|n, j|n} tau(lcm(i, j)).
a(n) = Sum_{d|n} tau_3(d^2) = Sum_{d|n} A007425(d^2). - Enrique Pérez Herrero, Jan 17 2013

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

A063182 Number of cyclic subgroups of the group S_n X S_n (where S_n is the symmetric group of degree n).

Original entry on oeis.org

1, 4, 26, 314, 5222, 168632, 5736908, 291993032, 18599068328, 1547379999392, 136254185631632, 18749419634845088, 2367416741670079712, 387737484226037810048, 78779133220155242489792, 17651532033334188604514432, 3945247307615376458903485568

Offset: 1

Vladeta Jovovic, Jul 10 2001

nonn

Cf. A051625, A060648, (unlabeled case) A063183.

a(9)-a(17) from Stephen A. Silver, Feb 22 2013

A308443 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(psi(k)/k), where psi() is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 1, 5, 23, 173, 1249, 13249, 130255, 1670297, 21350177, 322709021, 4933457671, 87302545285, 1551234590593, 30934738239833, 630934308253439, 14035903893341489, 320008164205036225, 7885477719156600757, 198735099970790861047, 5352424525748204265821

Offset: 0

Ilya Gutkovskiy, May 27 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..438

Cf. A000262, A001615, A060648, A156303.

nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^(DirichletConvolve[j, MoebiusMu[j]^2, j, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Sum[2^PrimeNu[d]/d, {d, Divisors[k]}] k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]

E.g.f.: exp(Sum_{k>=1} A060648(k)*x^k/k).

cp's OEIS Frontend

A203444 Numbers in range of Dedekind Psi function: A001615.

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A344222 a(n) = Sum_{k=1..n} tau(gcd(k,n)^4), where tau(n) is the number of divisors of n.

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A344304 Number of cyclic subgroups of the group (C_n)^8, where C_n is the cyclic group of order n.

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A344305 Number of cyclic subgroups of the group (C_n)^9, where C_n is the cyclic group of order n.

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A344306 Number of cyclic subgroups of the group (C_n)^10, where C_n is the cyclic group of order n.

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

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A062368 Multiplicative with a(p^e) = (e+1)*(e+2)*(4*e+3)/6.

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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).

A062368 Multiplicative with a(p^e) = (e+1)(e+2)(4*e+3)/6.