A060648 -id:A060648 - cp's OEIS frontend

A344219 Number of cyclic subgroups of the group (C_n)^5, where C_n is the cyclic group of order n.

1, 32, 122, 528, 782, 3904, 2802, 8464, 9923, 25024, 16106, 64416, 30942, 89664, 95404, 135440, 88742, 317536, 137562, 412896, 341844, 515392, 292562, 1032608, 488907, 990144, 803804, 1479456, 732542, 3052928, 954306, 2167056, 1964932, 2839744, 2191164, 5239344, 1926222

Offset: 1

Seiichi Manyama, May 12 2021

Inverse Moebius transform of A160893.

Seiichi Manyama, 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, A013663, A060648, A064969, A160893, A280184.

f[p_, e_] := 1 + ((p^5 - 1)/(p - 1))*((p^(4*e) - 1)/(p^4 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 15 2022 *)

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

a160893(n) = sumdiv(n, d, moebius(n/d)*d^5)/eulerphi(n);
a(n) = sumdiv(n, d, a160893(d));

a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n, x_5|n} phi(x_1)*phi(x_2)*phi(x_3)*phi(x_4)*phi(x_5)/phi(lcm(x_1, x_2, x_3, x_4, x_5)).
If p is prime, a(p) = 1 + (p^5 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^5 - 1)/(p - 1))*((p^(4*e) - 1)/(p^4 - 1)).
Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)/5) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 + 1/p^5) = 0.3939461744... . (End)

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)]

A344302 Number of cyclic subgroups of the group (C_n)^6, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 64, 365, 2080, 3907, 23360, 19609, 66592, 88817, 250048, 177157, 759200, 402235, 1254976, 1426055, 2130976, 1508599, 5684288, 2613661, 8126560, 7157285, 11338048, 6728905, 24306080, 12210157, 25743040, 21582653, 40786720, 21243691, 91267520, 29583457

Offset: 1

Seiichi Manyama, May 14 2021

Inverse Moebius transform of A160895.

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, A013664, A060648, A064969, A160895, A280184, A344219, A344303, A344304, A344305, A344306.

f[p_, e_] := 1 + ((p^6 - 1)/(p - 1))*((p^(5*e) - 1)/(p^5 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)

a160895(n) = sumdiv(n, d, moebius(n/d)*d^6)/eulerphi(n);
a(n) = sumdiv(n, d, a160895(d));

a(n) = Sum_{x_1|n, x_2|n, ..., x_6|n} phi(x_1)*phi(x_2)* ... *phi(x_6)/phi(lcm(x_1, x_2, ..., x_6)).
If p is prime, a(p) = 1 + (p^6 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^6 - 1)/(p - 1))*((p^(5*e) - 1)/(p^5 - 1)).
Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)/6) * Product_{p prime} ((1-1/p^5)/(p^2*(1-1/p))) = 0.32592074105... . (End)

A344303 Number of cyclic subgroups of the group (C_n)^7, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 128, 1094, 8256, 19532, 140032, 137258, 528448, 797891, 2500096, 1948718, 9032064, 5229044, 17569024, 21368008, 33820736, 25646168, 102130048, 49659542, 161256192, 150160252, 249435904, 154764794, 578122112, 305191407, 669317632, 581662904, 1133202048

Offset: 1

Seiichi Manyama, May 14 2021

Inverse Moebius transform of A160897.

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, A013665, A060648, A064969, A280184, A344219, A344302, A344304, A344305, A344306.

f[p_, e_] := 1 + ((p^7 - 1)/(p - 1))*((p^(6*e) - 1)/(p^6 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)

a160897(n) = sumdiv(n, d, moebius(n/d)*d^7)/eulerphi(n);
a(n) = sumdiv(n, d, a160897(d));

a(n) = Sum_{x_1|n, x_2|n, ..., x_7|n} phi(x_1)*phi(x_2)* ... *phi(x_7)/phi(lcm(x_1, x_2, ..., x_7)).
If p is prime, a(p) = 1 + (p^7 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^7 - 1)/(p - 1))*((p^(6*e) - 1)/(p^6 - 1)).
Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)/7) * Product_{p prime} ((1-1/p^6)/(p^2*(1-1/p))) = 0.2784611791... . (End)

A062367 Multiplicative with a(p^e) = (e+1)(e+2)(2*e+3)/6.

Original entry on oeis.org

1, 5, 5, 14, 5, 25, 5, 30, 14, 25, 5, 70, 5, 25, 25, 55, 5, 70, 5, 70, 25, 25, 5, 150, 14, 25, 30, 70, 5, 125, 5, 91, 25, 25, 25, 196, 5, 25, 25, 150, 5, 125, 5, 70, 70, 25, 5, 275, 14, 70, 25, 70, 5, 150, 25, 150, 25, 25, 5, 350, 5, 25, 70, 140, 25, 125, 5, 70, 25, 125, 5

Offset: 1

Vladeta Jovovic, Jul 07 2001

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A000012, A000330, A029939, A035116, A048691, A060648, A062368.

A062367 := proc(n)
    add(numtheory[tau](d)^2,d=numtheory[divisors](n)) ;
end proc:
seq(A062367(n),n=1..40) ; # R. J. Mathar, May 15 2025

{1}~Join~Array[Times @@ Map[((# + 1) (# + 2) (2 # + 3))/6 &, FactorInteger[#][[All, -1]] ] &, 70, 2] (* or *)
Array[DivisorSum[#, DivisorSigma[0, #]^2 &] &, 71] (* Michael De Vlieger, Mar 05 2021 *)

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

a(n) = Sum_{i|n, j|n} tau(gcd(i, j)) = Sum_{d|n} tau(d)^2.
a(n) = Sum_{i|n, j|n} tau(i)*tau(j)/tau(lcm(i, j)), where tau(n) = number of divisors of n, cf. A000005.
Dirichlet convolution of A035116 and A000012 (i.e., inverse Mobius transform of A035116). Dirichlet g.f.: zeta^5(s)/zeta(2s). - R. J. Mathar, Feb 03 2011
G.f.: Sum_{n>=1} A000005(n)^2*x^n/(1-x^n). - Mircea Merca, Feb 26 2014
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(tau(k)^2/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018
Dirichlet convolution of A007426 and A008966. Dirichlet convolution of A007425 and A034444. - R. J. Mathar, Jun 05 2020
Let b(n), n > 0, be Dirichlet inverse of a(n). Then b(n) is multiplicative with b(p^e) = (-1)^e*(Sum_{i=0..e} binomial(4,i)) for prime p and e >= 0, where binomial(n,k)=0 if n < k; abs(b(n)) is multiplicative and has the Dirichlet g.f.: (zeta(s))^5/(zeta(2*s))^4. - Werner Schulte, Feb 07 2021
a(n) = Sum_{d divides n} tau(d^2)*tau(n/d), Dirichlet convolution of A048691 and A000005. - Peter Bala, Jan 26 2024

A298473 a(n) = n * lambda(n) * 2^omega(n).

Original entry on oeis.org

1, -4, -6, 8, -10, 24, -14, -16, 18, 40, -22, -48, -26, 56, 60, 32, -34, -72, -38, -80, 84, 88, -46, 96, 50, 104, -54, -112, -58, -240, -62, -64, 132, 136, 140, 144, -74, 152, 156, 160, -82, -336, -86, -176, -180, 184, -94, -192, 98, -200, 204, -208, -106, 216, 220, 224, 228, 232, -118, 480

Offset: 1

Werner Schulte, Jan 19 2018

The sequence b(n) = abs(a(n)) = n * 2^omega(n) for n>=1 is multiplicative with b(p^e) = 2*p^e (p prime, e > 0) and is the Dirichlet inverse of a(n). The Dirichlet g.f. of b(n) is: (zeta(s-1))^2/zeta(2*s-2). For omega(n) and lambda(n) see A001221 and A008836, respectively.

			a(6) = a(2)*a(3) = (-4)*(-6) = 24 = 6*1*2^2;
a(8) = a(2^3) = 2*(-2)^3 = -16 = 8*(-1)*2^1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000290, A001221, A001222, A008683, A008836, A060648, A078439.

f:= proc(n) local t;
mul(2*(-t[1])^t[2],t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Mar 06 2022

Array[# (-1)^PrimeOmega[#]*2^PrimeNu[#] &, 60] (* Michael De Vlieger, Jan 20 2018 *)

a(n) = n*(-1)^bigomega(n)*2^omega(n); \\ Michel Marcus, Jan 20 2018

Multiplicative with a(p^e) = 2*(-p)^e (p prime, e>0).
Dirichlet inverse of abs(a(n)).
Dirichlet g.f.: zeta(2*s-2)/(zeta(s-1))^2.
Sum_{d|n} A000290(d)*a(n/d) = n*A060648(n).
Sum_{d|n} A078439(d)*a(n/d) = A008683(n).
O.g.f. for the unsigned sequence: Sum_{n >= 1} |a(n)|*x^n = Sum_{n >= 1} |mu(n)|*n*x^n/(1 - x^n)^2, where mu(n) = A008683(n) is the Möbius function. - Peter Bala, Mar 05 2022
Sum_{k=1..n} abs(a(k)) ~ 3*n^2/Pi^2 * (log(n) - 1/2 + 2*gamma - 12*zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 16 2025

A327251 Expansion of Sum_{k>=1} psi(k) * x^k / (1 - x^k)^2, where psi = A001615.

Original entry on oeis.org

1, 5, 7, 16, 11, 35, 15, 44, 33, 55, 23, 112, 27, 75, 77, 112, 35, 165, 39, 176, 105, 115, 47, 308, 85, 135, 135, 240, 59, 385, 63, 272, 161, 175, 165, 528, 75, 195, 189, 484, 83, 525, 87, 368, 363, 235, 95, 784, 161, 425, 245, 432, 107, 675, 253, 660, 273

Offset: 1

Ilya Gutkovskiy, Sep 15 2019

Inverse Moebius transform of A322577.

Dirichlet convolution of A001615 with A000027.

Cf. A000027, A001615, A018804, A060648, A322577, A347127.

nmax = 57; CoefficientList[Series[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k] x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := p^(e - 1)*((p + 1)*e + p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 24 2021 *)

mypsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = sumdiv(n, d, mypsi(n/d)*d); \\ Michel Marcus, Sep 15 2019

a(n) = Sum_{d|n} psi(n/d) * d.
a(p) = 2*p + 1, where p is prime.
Multiplicative with a(p^e) = p^(e-1)*((p+1)*e + p). - Antti Karttunen, Aug 24 2021

A344221 a(n) = Sum_{k=1..n} tau(gcd(k,n)^3), where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 5, 6, 13, 8, 30, 10, 29, 21, 40, 14, 78, 16, 50, 48, 61, 20, 105, 22, 104, 60, 70, 26, 174, 43, 80, 66, 130, 32, 240, 34, 125, 84, 100, 80, 273, 40, 110, 96, 232, 44, 300, 46, 182, 168, 130, 50, 366, 73, 215, 120, 208, 56, 330, 112, 290, 132, 160, 62, 624, 64, 170, 210, 253, 128, 420

Offset: 1

Seiichi Manyama, May 12 2021

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

Cf. A060648, A072691, A344222, A344223, A344224, A344225.

Table[Sum[DivisorSigma[0,GCD[k,n]^3],{k,n}],{n,100}] (* Giorgos Kalogeropoulos, May 13 2021 *)
f[p_, e_] := (p^e*(p + 2) - 3)/(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)^3));

a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^3));

PARI
```
a(n) = n*sumdiv(n, d, 3^omega(d)/d);
```

a(n) = Sum_{d|n} phi(n/d) * tau(d^3).
a(n) = n * Sum_{d|n} 3^omega(d) / d.
If p is prime, a(p) = 3 + p.
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = (p^e*(p + 2) - 3)/(p - 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 2/p^2) = 1.8019184198... . (End)

A062369 Dirichlet convolution of n and tau^2(n).

Original entry on oeis.org

1, 6, 7, 21, 9, 42, 11, 58, 30, 54, 15, 147, 17, 66, 63, 141, 21, 180, 23, 189, 77, 90, 27, 406, 54, 102, 106, 231, 33, 378, 35, 318, 105, 126, 99, 630, 41, 138, 119, 522, 45, 462, 47, 315, 270, 162, 51, 987, 86, 324, 147, 357, 57, 636, 135, 638, 161, 198, 63, 1323

Offset: 1

Vladeta Jovovic, Jul 07 2001

Dirichlet convolution of A000027 and A035116.

Inverse Mobius transform of A060724. - R. J. Mathar, Oct 15 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000203, A060648.

Cf. A000005, A060724, A064950, A062369, A062368, A062380.

[&+[d*#Divisors(Floor(n/d))^2:d in Divisors(n)]:n in [1..60]]; // Marius A. Burtea, Aug 25 2019

a[n_] := Sum[ DivisorSigma[1, i]*DivisorSigma[1, j] / DivisorSigma[1, LCM[i, j]], {i, Divisors[n]}, {j, Divisors[n]}]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 26 2013 *)

a(n) = sumdiv(n, d, d*numdiv(n/d)^2); \\ Michel Marcus, Nov 03 2018

a(n) = Sum_{i|n, j|n} sigma(i)*sigma(j)/sigma(lcm(i,j)), where sigma(n) = sum of divisors of n.
a(n) = Sum_{i|d, j|d} sigma(gcd(i, j));
a(n) = Sum_{d|n} d*tau(n/d)^2, where tau(n) = number of divisors of n.
Multiplicative with a(p^e) = (1-p^(3+e)-p^(2+e)+e^2+4*p^2+p^2*e^2+2*e-3*p+4*p^2*e-6*e*p-2*e^2*p)/(1-p)^3.
Dirichlet g.f.: (zeta(s))^4*zeta(s-1)/zeta(2*s). - R. J. Mathar, Feb 09 2011
G.f.: Sum_{k>=1} tau(k)^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018
Sum_{k=1..n} a(k) ~ 5 * Pi^4 * n^2 / 144. - Vaclav Kotesovec, Jan 28 2019
a(n) = Sum_{d|n} tau(d^2)*sigma(n/d), where tau(n) = number of divisors of n, and sigma(n) = sum of divisors of n. - Ridouane Oudra, Aug 25 2019

cp's OEIS Frontend

A344219 Number of cyclic subgroups of the group (C_n)^5, where C_n is the cyclic group of order n.

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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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A344302 Number of cyclic subgroups of the group (C_n)^6, where C_n is the cyclic group of order n.

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A344303 Number of cyclic subgroups of the group (C_n)^7, where C_n is the cyclic group of order n.

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PARI

Formula

A062367 Multiplicative with a(p^e) = (e+1)*(e+2)*(2*e+3)/6.

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Maple

Mathematica

PARI

Formula

A298473 a(n) = n * lambda(n) * 2^omega(n).

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Maple

A062367 Multiplicative with a(p^e) = (e+1)(e+2)(2*e+3)/6.