A216624 -id:A216624 - 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

A216626 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} lcm(c,d) for n>=1, k>=1.

Original entry on oeis.org

1, 3, 3, 4, 7, 4, 7, 12, 12, 7, 6, 15, 10, 15, 6, 12, 18, 28, 28, 18, 12, 8, 28, 24, 27, 24, 28, 8, 15, 24, 30, 42, 42, 30, 24, 15, 13, 31, 32, 60, 16, 60, 32, 31, 13, 18, 39, 60, 56, 72, 72, 56, 60, 39, 18, 12, 42, 28, 51, 48, 70, 48, 51, 28, 42, 12, 28, 36

Offset: 1

Peter Luschny, Sep 12 2012

T(n,n) = A064950(n) = sum_{d|n} d*tau(d^2).
T(n,1) = T(1,n) = A000203(n) = sigma(n).
T(n,2) = T(2,n) = A062731(n) = sigma(2*n).
T(n+1,n) = A083539(n) = sigma(n+1)*sigma(n).
T(prime(n),1) = A008864(n) = prime(n)+1.

			[-----1---2---3----4----5----6----7----8----9---10---11---12]
[ 1]  1,  3,  4,   7,   6,  12,   8,  15,  13,  18,  12,  28
[ 2]  3,  7, 12,  15,  18,  28,  24,  31,  39,  42,  36,  60
[ 3]  4, 12, 10,  28,  24,  30,  32,  60,  28,  72,  48,  70
[ 4]  7, 15, 28,  27,  42,  60,  56,  51,  91,  90,  84, 108
[ 5]  6, 18, 24,  42,  16,  72,  48,  90,  78,  48,  72, 168
[ 6] 12, 28, 30,  60,  72,  70,  96, 124,  84, 168, 144, 150
[ 7]  8, 24, 32,  56,  48,  96,  22, 120, 104, 144,  96, 224
[ 8] 15, 31, 60,  51,  90, 124, 120,  83, 195, 186, 180, 204
[ 9] 13, 39, 28,  91,  78,  84, 104, 195,  55, 234, 156, 196
[10] 18, 42, 72,  90,  48, 168, 144, 186, 234, 112, 216, 360
[11] 12, 36, 48,  84,  72, 144,  96, 180, 156, 216,  34, 336
[12] 28, 60, 70, 108, 168, 150, 224, 204, 196, 360, 336, 270
.
Displayed as a triangular array:
    1;
    3,  3;
    4,  7,  4;
    7, 12, 12,  7;
    6, 15, 10, 15,  6;
   12, 18, 28, 28, 18, 12;
    8, 28, 24, 27, 24, 28,  8;
   15, 24, 30, 42, 42, 30, 24, 15;
   13, 31, 32, 60, 16, 60, 32, 31, 13;

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Cf. A216620, A216621, A216622, A216623, A216624, A216625, A216627.

with(numtheory):
T:= (n, k) -> add(add(ilcm(c, d), c=divisors(n)), d=divisors(k)):
seq (seq (T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 12 2012

T[n_, k_] := Sum[LCM[c, d], {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2014 *)

def A216626(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(lcm, cp))
for n in (1..12): [A216626(n,k) for k in (1..12)]

A216620 Square array read by antidiagonals: T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)) for n>=1, k>=1.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 3, 4, 4, 3, 2, 6, 5, 6, 2, 4, 4, 6, 6, 4, 4, 2, 8, 4, 10, 4, 8, 2, 4, 4, 10, 6, 6, 10, 4, 4, 3, 8, 4, 12, 7, 12, 4, 8, 3, 4, 6, 8, 6, 8, 8, 6, 8, 6, 4, 2, 8, 8, 14, 4, 20, 4, 14, 8, 8, 2, 6, 4, 8, 9, 8, 8, 8, 8, 9, 8, 4, 6, 2, 12, 4, 12, 6

Offset: 1

Peter Luschny, Sep 12 2012

T(n,n) = A060648(n) = Sum_{d|n} Dedekind_Psi(d).
T(n,1) = T(1,n) = A000005(n) = tau(n).
T(n,2) = T(2,n) = A062011(n) = 2*tau(n).
T(n+1,n) = A092517(n) = tau(n+1)*tau(n).
T(prime(n),1) = A007395(n) = 2.
T(prime(n),prime(n)) = A052147(n) = prime(n)+2.

			[----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,  4,  4,  6,  4,  8,  4,  8,  6,  8,  4, 12
[ 3] 2,  4,  5,  6,  4, 10,  4,  8,  8,  8,  4, 15
[ 4] 3,  6,  6, 10,  6, 12,  6, 14,  9, 12,  6, 20
[ 5] 2,  4,  4,  6,  7,  8,  4,  8,  6, 14,  4, 12
[ 6] 4,  8, 10, 12,  8, 20,  8, 16, 16, 16,  8, 30
[ 7] 2,  4,  4,  6,  4,  8,  9,  8,  6,  8,  4, 12
[ 8] 4,  8,  8, 14,  8, 16,  8, 22, 12, 16,  8, 28
[ 9] 3,  6,  8,  9,  6, 16,  6, 12, 17, 12,  6, 24
[10] 4,  8,  8, 12, 14, 16,  8, 16, 12, 28,  8, 24
[11] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 13, 12
[12] 6, 12, 15, 20, 12, 30, 12, 28, 24, 24, 12, 50
.
Displayed as a triangular array:
   1,
   2, 2,
   2, 4,  2,
   3, 4,  4,  3,
   2, 6,  5,  6, 2,
   4, 4,  6,  6, 4,  4,
   2, 8,  4, 10, 4,  8, 2,
   4, 4, 10,  6, 6, 10, 4, 4,
   3, 8,  4, 12, 7, 12, 4, 8, 3,

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Cf. A216621, A216622, A216623, A216624, A216625, A216626, A216627.

with(numtheory):
T:= (n, k)-> add(add(phi(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[n_, k_] := Outer[ EulerPhi[ GCD[#1, #2]]&, Divisors[n], Divisors[k]] // Flatten // Total; Table[ t[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *)

def A216620(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(euler_phi, map(gcd, cp)))
for n in (1..12): [A216620(n,k) for k in (1..12)]

A216621 Triangle read by rows, n >= 1, 1 <= k <= n, T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)).

Original entry on oeis.org

1, 2, 4, 2, 4, 5, 3, 6, 6, 10, 2, 4, 4, 6, 7, 4, 8, 10, 12, 8, 20, 2, 4, 4, 6, 4, 8, 9, 4, 8, 8, 14, 8, 16, 8, 22, 3, 6, 8, 9, 6, 16, 6, 12, 17, 4, 8, 8, 12, 14, 16, 8, 16, 12, 28, 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 13, 6, 12, 15, 20, 12, 30, 12, 28, 24, 24, 12

Offset: 1

Peter Luschny, Sep 12 2012

This is the lower triangular array of A216620, which is the main entry for this sequence.
T(n,1) = A000005(n) = tau(n).
T(n,n) = A060648(n) = sum{d|n} Dedekind_Psi(d).

			The first rows of the triangle are:
  1;
  2,  4;
  2,  4,  5;
  3,  6,  6, 10;
  2,  4,  4,  6,  7;
  4,  8, 10, 12,  8, 20;
  2,  4,  4,  6,  4,  8,  9;
  4,  8,  8, 14,  8, 16,  8, 22;
  3,  6,  8,  9,  6, 16,  6, 12, 17;
  4,  8,  8, 12, 14, 16,  8, 16, 12, 28;
  2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 13;

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A216620, A216622, A216623, A216624, A216625, A216626, A216627.

with(numtheory):
T:= (n, k)-> add(add(phi(igcd(c,d)), c=divisors(n)), d=divisors(k)):
seq (seq (T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Sep 12 2012

t[n_, k_] := Sum[ EulerPhi[GCD[c, d]], {c, Divisors[n]}, {d, Divisors[k]}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

for n in (1..9): [A216620(n,k) for k in (1..n)]

A216622 Square array read by antidiagonals: T(n,k) = Sum_{c|n, d|k} phi(lcm(c,d)) for n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 7, 8, 5, 6, 10, 12, 12, 10, 6, 7, 12, 15, 14, 15, 12, 7, 8, 14, 14, 20, 20, 14, 14, 8, 9, 16, 21, 24, 13, 24, 21, 16, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 20, 19, 26, 35, 28, 35, 26, 19, 20, 11, 12, 22, 30, 36, 40

Offset: 1

Peter Luschny, Sep 12 2012

T(n,n) = A062380(n) = Sum_{d|n} phi(d)*tau(d^2).
T(n,1) = T(1,n) = A000027(n) = n.
T(n,2) = T(2,n) = A005843(n) = 2*n.
T(n+1,n) = A002378(n) = (n+1)*n.
T(prime(n),1) = A000040(n) = prime(n).
T(prime(n),prime(n)) = 3*prime(n)-2.

			[-----1---2---3---4---5---6---7---8---9---10---11---12]
[ 1]  1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12
[ 2]  2,  4,  6,  8, 10, 12, 14, 16, 18,  20,  22,  24
[ 3]  3,  6,  7, 12, 15, 14, 21, 24, 19,  30,  33,  28
[ 4]  4,  8, 12, 14, 20, 24, 28, 26, 36,  40,  44,  42
[ 5]  5, 10, 15, 20, 13, 30, 35, 40, 45,  26,  55,  60
[ 6]  6, 12, 14, 24, 30, 28, 42, 48, 38,  60,  66,  56
[ 7]  7, 14, 21, 28, 35, 42, 19, 56, 63,  70,  77,  84
[ 8]  8, 16, 24, 26, 40, 48, 56, 42, 72,  80,  88,  78
[ 9]  9, 18, 19, 36, 45, 38, 63, 72, 37,  90,  99,  76
[10] 10, 20, 30, 40, 26, 60, 70, 80, 90,  52, 110, 120
[11] 11, 22, 33, 44, 55, 66, 77, 88, 99, 110,  31, 132
[12] 12, 24, 28, 42, 60, 56, 84, 78, 76, 120, 132,  98
.
Displayed as a triangular array:
   1,
   2,  2,
   3,  4,  3,
   4,  6,  6,  4,
   5,  8,  7,  8,  5,
   6, 10, 12, 12, 10,  6,
   7, 12, 15, 14, 15, 12,  7,
   8, 14, 14, 20, 20, 14, 14,  8,
   9, 16, 21, 24, 13, 24, 21, 16,  9,

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Cf. A216620, A216621, A216623, A216624, A216625, A216626, A216627.

with(numtheory):
T:= (n, k)-> add(add(phi(ilcm(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[n_, k_] := Sum[ EulerPhi[LCM[c, d]], {c, Divisors[n]}, {d, Divisors[k]}]; Table[ t[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

def A216622(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(euler_phi, map(lcm, cp)))
for n in (1..12): [A216622(n,k) for k in (1..12)]

A216623 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = Sum_{c|n,d|k} phi(lcm(c,d)).

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 4, 8, 12, 14, 5, 10, 15, 20, 13, 6, 12, 14, 24, 30, 28, 7, 14, 21, 28, 35, 42, 19, 8, 16, 24, 26, 40, 48, 56, 42, 9, 18, 19, 36, 45, 38, 63, 72, 37, 10, 20, 30, 40, 26, 60, 70, 80, 90, 52, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 31, 12, 24

Offset: 1

Peter Luschny, Sep 12 2012

This is the lower triangular array of A216622, which is the main entry for this sequence.
T(n,1) = A000027(n).
T(n,n) = A062380(n).

			The first rows of the triangle are:
1,
2,  4,
3,  6,  7,
4,  8, 12, 14,
5, 10, 15, 20, 13,
6, 12, 14, 24, 30, 28,
7, 14, 21, 28, 35, 42, 19,
8, 16, 24, 26, 40, 48, 56, 42,
9, 18, 19, 36, 45, 38, 63, 72, 37,

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A216620, A216621, A216622, A216624, A216625, A216626, A216627.

with(numtheory):
T:= (n, k)-> add(add(phi(ilcm(c, d)), c=divisors(n)), d=divisors(k)):
seq (seq (T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Sep 12 2012

t[n_, k_] := Sum[ EulerPhi[ LCM[c, d]], {c, Divisors[n]}, {d, Divisors[k]}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 23 2013 *)

# uses[A216622]
for n in (1..9): [A216622(n,k) for k in (1..n)]

A216625 Triangle read by rows, n >= 1, 1 <= k <= n, T(n,k) = Sum_{c|n,d|k} gcd(c,d).

Original entry on oeis.org

1, 2, 5, 2, 4, 6, 3, 8, 6, 15, 2, 4, 4, 6, 8, 4, 10, 12, 16, 8, 30, 2, 4, 4, 6, 4, 8, 10, 4, 11, 8, 22, 8, 22, 8, 37, 3, 6, 10, 9, 6, 20, 6, 12, 23, 4, 10, 8, 16, 16, 20, 8, 22, 12, 40, 2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 14, 6, 16, 18, 30, 12, 48, 12, 44, 30, 32

Offset: 1

Peter Luschny, Sep 12 2012

This is the lower triangular array of A216624, which is the main entry for this sequence.
T(n,1) = A000005(n) = tau(n).
T(n,n) = A060724(n) = Sum_{d|n} d*tau((n/d)^2).

			The first rows of the triangle are:
  1;
  2,  5;
  2,  4,  6;
  3,  8,  6, 15;
  2,  4,  4,  6,  8;
  4, 10, 12, 16,  8, 30;
  2,  4,  4,  6,  4,  8, 10;
  4, 11,  8, 22,  8, 22,  8, 37;
  3,  6, 10,  9,  6, 20,  6, 12, 23;

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A216620, A216621, A216622, A216623, A216624, A216626, A216627.

with(numtheory):
T:= (n, k)-> add(add(igcd(c, d), c=divisors(n)), d=divisors(k)):
seq (seq (T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Sep 12 2012

T[n_, k_] := Sum[GCD[c, d], {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2014 *)

for n in (1..9): [A216624(n,k) for k in (1..n)]

A216627 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = sum_{c|n,d|k} lcm(c,d).

Original entry on oeis.org

1, 3, 7, 4, 12, 10, 7, 15, 28, 27, 6, 18, 24, 42, 16, 12, 28, 30, 60, 72, 70, 8, 24, 32, 56, 48, 96, 22, 15, 31, 60, 51, 90, 124, 120, 83, 13, 39, 28, 91, 78, 84, 104, 195, 55, 18, 42, 72, 90, 48, 168, 144, 186, 234, 112, 12, 36, 48, 84, 72, 144, 96, 180, 156

Offset: 1

Peter Luschny, Sep 12 2012

This is the lower triangular array of A216626, which is the main entry for this sequence.

			The first rows of the triangle are:
1;
3,   7;
4,  12, 10;
7,  15, 28, 27;
6,  18, 24, 42, 16;
12, 28, 30, 60, 72,  70;
8,  24, 32, 56, 48,  96,  22;
15, 31, 60, 51, 90, 124, 120,  83;
13, 39, 28, 91, 78,  84, 104, 195, 55;

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A216620, A216621, A216622, A216623, A216624, A216625, A216626.

with(numtheory):
T:= (n, k) -> add(add(ilcm(c, d), c=divisors(n)), d=divisors(k));
seq (seq (T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Sep 12 2012

T[n_, k_] := Sum[LCM[c, d], {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2014 *)

for n in (1..9): [A216626(n,k) for k in (1..n)]

T(n,1) = A000203(n) = sigma(n).

T(n,n) = A064950(n) = sum_{d|n} d*tau(d^2).

A060710 Number of subgroups of dihedral group with 2n elements, counting conjugate subgroups only once, i.e., conjugacy classes of subgroups of the dihedral group.

Original entry on oeis.org

2, 5, 4, 8, 4, 10, 4, 11, 6, 10, 4, 16, 4, 10, 8, 14, 4, 15, 4, 16, 8, 10, 4, 22, 6, 10, 8, 16, 4, 20, 4, 17, 8, 10, 8, 24, 4, 10, 8, 22, 4, 20, 4, 16, 12, 10, 4, 28, 6, 15, 8, 16, 4, 20, 8, 22, 8, 10, 4, 32, 4, 10, 12, 20, 8, 20, 4, 16, 8, 20, 4, 33, 4, 10, 12, 16, 8, 20, 4, 28, 10, 10, 4

Offset: 1

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

nonn

The total number of subgroups, counting conjugate subgroups as distinct, is A007503.

Also the number of subgroups of the group C_n x C_2 (where C_n is the cyclic group with n elements).

			The dihedral group D6 is isomorphic to the symmetric group S_3 and the conjugacy classes of subgroups are: the trivial group, the whole group, subgroup of order 2 generated by a transposition and the subgroup A_3 generated by the 3-cycles. So a(3) = 4.

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

Cf. A001622, A007503, A062011.

A row of A216624.

a[n_] := DivisorSum[n, 3-Mod[#,2]&];
Array[a, 100] (* Jean-François Alcover, Jun 03 2019 *)

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

{ for (n=1, 1000, write("b060710.txt", n, " ", sumdiv(n, d, 3 - d%2)); ) } \\ Harry J. Smith, Jul 10 2009

def A060710(n): return add(3 - int(is_odd(d)) for d in divisors(n))
[A060710(n) for n in (1..83)] # Peter Luschny, Sep 12 2012

For even n, a(n) = 2*tau(n) + tau(n/2).
For odd n, a(n) = tau(2n) = 2*tau(n) = 2*A000005(n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001
From Michael Somos, Sep 20 2005: (Start)
Moebius transform is period 2 sequence [2, 3, ...].
G.f.: Sum_{k>0} x^k*(2+3x^k)/(1-x^(2k)) = Sum_{k>0} 2*x^(2k-1)/(1-x^(2k-1)) + 3*x^(2k)/(1-x^(2k)). (End)
a(n) = 4*tau(n) - tau(2n). - Ridouane Oudra, Jan 16 2023
Sum_{k=1..n} a(k) ~ n*(5*log(n) + 10*gamma - log(2) - 5)/2, where gamma is Euler's constant (A001622). - Amiram Eldar, Jan 21 2023

More terms from Vladeta Jovovic, Jul 15 2001

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

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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A216626 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} lcm(c,d) for n>=1, k>=1.

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A216620 Square array read by antidiagonals: T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)) for n>=1, k>=1.

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A216621 Triangle read by rows, n >= 1, 1 <= k <= n, T(n,k) = Sum_{c|n,d|k} phi(gcd(c,d)).

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A216622 Square array read by antidiagonals: T(n,k) = Sum_{c|n, d|k} phi(lcm(c,d)) for n >= 1, k >= 1.

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A216623 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = Sum_{c|n,d|k} phi(lcm(c,d)).

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A216625 Triangle read by rows, n >= 1, 1 <= k <= n, T(n,k) = Sum_{c|n,d|k} gcd(c,d).

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