A054584 -id:A054584 - cp's OEIS frontend

A060724 Number of subgroups of the group C_n X C_n (where C_n is the cyclic group of order n).

1, 5, 6, 15, 8, 30, 10, 37, 23, 40, 14, 90, 16, 50, 48, 83, 20, 115, 22, 120, 60, 70, 26, 222, 45, 80, 76, 150, 32, 240, 34, 177, 84, 100, 80, 345, 40, 110, 96, 296, 44, 300, 46, 210, 184, 130, 50, 498, 75, 225, 120, 240, 56, 380, 112, 370, 132, 160, 62, 720, 64

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001

			a(2) = 5 because for the group C_2 X C_2 there are the following subgroups: the trivial subgroup, the whole group and the three subgroups of order 2.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, On the subgroups of the group Z_m X Z_n, arXiv preprint arXiv:1211.1797 [math.GR], 2012. - From _N. J. A. Sloane_, Jan 02 2013
W. G. Nowak and L. Tóth, On the average number of subgroups of the group Z_m X Z_n, arXiv preprint arXiv:1307.1414 [math.NT], 2013.
Laszlo Toth, On the number of cyclic subgroups of a finite abelian group, arXiv preprint arXiv:1203.6201 [math.GR], 2012. - From _N. J. A. Sloane_, Sep 22 2012
L. Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
Index entries for sequences related to groups

Cf. A060648, A050488, A054584, A000005, A062369, A344132, A344138, A344139, A344140.

Main diagonal of A216624.

List([1..50], n->Sum(ConjugacyClassesSubgroups( LatticeSubgroups( DirectProduct( List([n, n], k->CyclicGroup(k)) ))), Size)); # Andrew Howroyd, Jul 01 2018

for n from 1 to 200 do: ans := 1: for i from 1 to nops(ifactors(n)[2]) do p := ifactors(n)[2][i][1]: e := ifactors(n)[2][i][2]: ans := ans*(p^(e+2)+p^(e+1)+1+2*e-3*p-2*e*p)/(p-1)^2: od: printf(`%d,`,ans): od:

ppCase[ {p_Integer, e_Integer} ] := (1-2*e*(p-1)+p*(p^e*(1+p)-3))/(p-1)^2; Table[ Times @@ (ppCase /@ FactorInteger[ i ]), {i, 1, 100} ]

a(n)={sumdiv(n, d, eulerphi(n/d)*numdiv(d)^2)} \\ Andrew Howroyd, Jul 01 2018

a(n) = sum(k=1, n, numdiv(gcd(k, n))^2); \\ Seiichi Manyama, May 11 2021

def A060724(n) :
    d = divisors(n); cp = cartesian_product([d, d])
    return reduce(lambda x,y: x+y, map(gcd, cp))
[A060724(n) for n in (1..61)]   # Peter Luschny, Sep 10 2012

a(n) is multiplicative: if the canonical factorization of n is the product of p^e(p) over primes then a(n) = product a(p^e(p)). For a prime p: a(p) = p + 3.
a(p^e) = (p^(e+2)+p^(e+1)+1+2*e-3*p-2*e*p)/(p-1)^2.
a(n) = Sum_{i|n, j|n} gcd(i, j). - Vladeta Jovovic, Oct 28 2001
Also a(n) = Sum_{d|n} d*tau((n/d)^2). - Vladeta Jovovic, Apr 01 2002
Also a(n) = Sum_{d|n} phi(n/d)*tau(d)^2.
Inverse Moebius transform of A060648. - Vladeta Jovovic, Mar 31 2009
Dirichlet g.f. zeta^3(s)*zeta(s-1)/zeta(2*s). - R. J. Mathar, Mar 14 2011
a(n) = Sum_{d|n} psi(d)*tau(n/d), where psi is A001615 and tau is A000005. - Enrique Pérez Herrero, Feb 29 2012
Sum_{k=1..n} a(k) ~ 5 * Pi^2 * n^2 / 24. - Vaclav Kotesovec, Jun 02 2019
a(n) = Sum_{k=1..n} tau(gcd(k,n))^2. - Seiichi Manyama, May 11 2021

Formula and more terms from Vladeta Jovovic, Jul 06 2001

A035191 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 9.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 2, 4, 1, 4, 2, 3, 2, 4, 2, 5, 2, 2, 2, 6, 2, 4, 2, 4, 3, 4, 1, 6, 2, 4, 2, 6, 2, 4, 4, 3, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 5, 3, 6, 2, 6, 2, 2, 4, 8, 2, 4, 2, 6, 2, 4, 2, 7, 4, 4, 2, 6, 2, 8, 2, 4, 2, 4, 3, 6, 4, 4, 2, 10, 1

Offset: 1

N. J. A. Sloane

Number of divisors of n not congruent to 0 mod 3. - Vladeta Jovovic, Oct 26 2001
a(n) is the number of factors (over Q) of the polynomial x^(2n) + x^n + 1 . a(n) = d(3n) - d(n) where d() is the divisor function. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003
Equals Mobius transform of A011655. - Gary W. Adamson, Apr 24 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A011655, A035207, A046913, A054584.

Cf. A000005, A001227, A001620, A001817, A001817, A038502.

a035191 n = a001817 n + a001822 n  -- Reinhard Zumkeller, Nov 26 2011

[NumberOfDivisors(n)/Valuation(3*n, 3): n in [1..100]]; // Vincenzo Librandi, Jun 03 2019

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 := 1:else b := e+1:fi:s := s*b:od:printf(`%d,`,s); od:
# alternative
A035191 := proc(n)
    A001817(n)+A001822(n) ;
end proc:
[seq(A035191(n),n=1..100)] ; # R. J. Mathar, Sep 25 2017

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[9, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
f[3, e_] := 1; f[p_, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)

my(m=9); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(9, d)); \\ Amiram Eldar, Nov 20 2023

Multiplicative with a(3^e)=1 and a(p^e)=e+1 for p<>3.
G.f.: Sum_{k>0} x^k*(1+x^k)/(1-x^(3*k)). - Vladeta Jovovic, Dec 16 2002
a(n) = A001817(n) + A001822(n). [Reinhard Zumkeller, Nov 26 2011]
a(n) = tau(3*n) - tau(n). - Ridouane Oudra, Sep 05 2020
From Amiram Eldar, Nov 27 2022: (Start)
Dirichlet g.f.: zeta(s)^2 * (1 - 1/3^s).
Sum_{k=1..n} a(k) ~ (2*n*log(n) + (4*gamma + log(3) - 2)*n)/3, where gamma is Euler's constant (A001620). (End)
a(n) = Sum_{d|n} Kronecker(9, d). - Amiram Eldar, Nov 20 2023
a(n) = A000005(A038502(n)). - Ridouane Oudra, Sep 30 2024

A216624 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} gcd(c,d) for n>=1, k>=1.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 3, 4, 4, 3, 2, 8, 6, 8, 2, 4, 4, 6, 6, 4, 4, 2, 10, 4, 15, 4, 10, 2, 4, 4, 12, 6, 6, 12, 4, 4, 3, 11, 4, 16, 8, 16, 4, 11, 3, 4, 6, 8, 6, 8, 8, 6, 8, 6, 4, 2, 10, 10, 22, 4, 30, 4, 22, 10, 10, 2, 6, 4, 8, 9, 8, 8, 8, 8, 9, 8, 4, 6

Offset: 1

Peter Luschny, Sep 12 2012

T(n,k) = number of subgroups of C_n X C_k. [Hampjes et al.] - N. J. A. Sloane, Feb 02 2013

			[----1---2---3---4---5---6---7---8---9--10--11--12]
[ 1] 1,  2,  2,  3,  2,  4,  2,  4,  3,  4,  2,  6
[ 2] 2,  5,  4,  8,  4, 10,  4, 11,  6, 10,  4, 16
[ 3] 2,  4,  6,  6,  4, 12,  4,  8, 10,  8,  4, 18
[ 4] 3,  8,  6, 15,  6, 16,  6, 22,  9, 16,  6, 30
[ 5] 2,  4,  4,  6,  8,  8,  4,  8,  6, 16,  4, 12
[ 6] 4, 10, 12, 16,  8, 30,  8, 22, 20, 20,  8, 48
[ 7] 2,  4,  4,  6,  4,  8, 10,  8,  6,  8,  4, 12
[ 8] 4, 11,  8, 22,  8, 22,  8, 37, 12, 22,  8, 44
[ 9] 3,  6, 10,  9,  6, 20,  6, 12, 23, 12,  6, 30
[10] 4, 10,  8, 16, 16, 20,  8, 22, 12, 40,  8, 32
[11] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 14, 12
[12] 6, 16, 18, 30, 12, 48, 12, 44, 30, 32, 12, 90
.
Displayed as a triangular array:
1,
2,  2,
2,  5,  2,
3,  4,  4,  3,
2,  8,  6,  8, 2,
4,  4,  6,  6, 4,  4,
2, 10,  4, 15, 4, 10, 2,
4,  4, 12,  6, 6, 12, 4,  4,
3, 11,  4, 16, 8, 16, 4, 11, 3,

Alois P. Heinz, Antidiagonals n = 1..141, flattened
M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, On the subgroups of the group Z_m X Z_n, 2012. - From _N. J. A. Sloane_, Feb 02 2013

Cf. A216620, A216621, A216622, A216623, A216625, A216626, A216627.
Main diagonal is A060724.
Rows give A000005, A060710, A054584, A221951, A221852.

with(numtheory):
T:= (n, k)-> add(add(igcd(c,d), c=divisors(n)), d=divisors(k)):
seq(seq(T(n, 1+d-n), n=1..d), d=1..14); # Alois P. Heinz, Sep 12 2012
T:=proc(m,n) local d; add( d*tau(m*n/d^2), d in divisors(gcd(m,n))); end; # N. J. A. Sloane, Feb 02 2013

t[n_, k_] := Sum[Sum[GCD[c, d], {c, Divisors[n]}], {d, Divisors[k]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 21 2013 *)

def A216624(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(gcd, cp))
for n in (1..12): [A216624(n,k) for k in (1..12)]

T(n,n) = A060724(n) = sum_{d|n} d*tau((n/d)^2).
T(n,1) = T(1,n) = A000005(n) = tau(n).
T(n,2) = T(2,n) = A060710(n) = sum_{d|n} (3-[d is odd]) (Iverson bracket).
T(n+1,n) = A092517(n) = tau(n+1)*tau(n).
T(prime(n),1) = A007395(n) = 2.
T(prime(n),prime(n)) = A113935(n) = prime(n)+3.

cp's OEIS Frontend

A060724 Number of subgroups of the group C_n X C_n (where C_n is the cyclic group of order n).

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A035191 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 9.

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A216624 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} gcd(c,d) for n>=1, k>=1.

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A035191 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 9.