A060724 -id:A060724 - cp's OEIS frontend

A060648 Number of cyclic subgroups of the group C_n X C_n (where C_n is the cyclic group of order n).

1, 4, 5, 10, 7, 20, 9, 22, 17, 28, 13, 50, 15, 36, 35, 46, 19, 68, 21, 70, 45, 52, 25, 110, 37, 60, 53, 90, 31, 140, 33, 94, 65, 76, 63, 170, 39, 84, 75, 154, 43, 180, 45, 130, 119, 100, 49, 230, 65, 148, 95, 150, 55, 212, 91, 198, 105, 124, 61, 350, 63, 132, 153, 190

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 04 2001

The group U(n) of units modulo n acts on the direct product (Z_n)^k by multiplication. The number g(n,k) of orbits of U(n) acting on Z/(n)^k is g(n,k) = (1/phi(n))*Sum(gcd(n,a-1)^k) where the sum is over a in U(n) and phi(n) is the Euler totient function. A060648 gives g(n,2). - W. Edwin Clark, Jul 20 2001

a(n) is also the number of orbits of length n for the map TxT (Cartesion product) where T is a map with one orbit of each length. - Thomas Ward, Apr 08 2009

			The cycle index of C_4 X C_4 is (x(1)^4 + x(2)^2 + 2*x(4))^2 = x(1)^8 + 2*x(1)^4*x(2)^2 + 4*x(1)^4*x(4) + x(2)^4 + 4*x(2)^2*x(4) + 4*x(4)^2 and C_4 X C_4 has 1 element of order 1, 3 elements of order 2 and 12 elements of order 4. So a(4) = 1/phi(1) + 3/phi(2) + 12/phi(4) = 10, where phi = Euler totient function, cf. A000010. - _Vladeta Jovovic_, Jul 05 2001
For a(4) the pairs (e,d) are (1,4),(2,4),(4,4),(4,2),(4,1) with gcds 1,2,4,2,1 resp. giving 10 in total. - _Thomas Ward_, Apr 08 2009

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], 2012-2014. - 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.
Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, Journal of Integer Sequences, 12 (2009) Article 09.2.4.
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110, and arXiv:1103.5861, [math.NT], 2011.
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013-2014.
Index entries for sequences related to groups.

Cf. A060724, A063379, A061503, A064969, A216620, A280184, A344219, A344302, A344303.

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+1)+p^e-2)/(p-1):od:printf(`%d,`,ans):od:

Table[ Plus @@ Map[ Times @@ (EulerPhi /@ #)/EulerPhi[ LCM @@ # ] &, Flatten[ Outer[ {##} &, Divisors[ i ], Divisors[ i ] ], 1 ] ], {i, 1, 100} ]
f[p_, e_] := (p^(e+1)+p^e-2)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 20 2020 *)

a(n) = sumdiv(n, d,  2^omega(d)*(n/d) ); \\ Joerg Arndt, Sep 16 2012

def A060648(n) :
    def dedekind_psi(n) : return n*mul(1+1/p for p in prime_divisors(n))
    return reduce(lambda x,y: x+y, [dedekind_psi(d) for d in divisors(n)])
[A060648(n) for n in (1..64)]  # 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)). If n = p^e, p prime, a(n) = (p^(e+1)+p^e-2)/(p-1).
a(n) = Sum_{i|n, j|n} phi(i)*phi(j)/phi(lcm(i, j)). - Vladeta Jovovic, Jul 07 2001
a(n) = Sum_{i|n, j|n} phi(gcd(i, j)).
a(n) = Sum_{d|n} phi(n/d)*tau(d^2).
a(n) = sum(d|n, sigma(d)*moebius(n/d)^2 ). - Benoit Cloitre, Sep 08 2002
Inverse Euler transform of A156302. - Vladeta Jovovic, Feb 14 2009
Moebius transform of A060724. - Vladeta Jovovic, Apr 05 2009
Also a(n) = (1/n)*Sum_{d|n} sigma(d)^2*moebius(n/d). - Vladeta Jovovic, Mar 31 2009
Inverse Moebius transform of A001615. - Vladeta Jovovic, Apr 05 2009
From Thomas Ward, Apr 08 2009: (Start)
a(n) = Sum_{lcm(e,d)=n} gcd(e,d).
Dirichlet g.f.: zeta(s)^2*zeta(s-1)/zeta(2s). (End)
For the proofs of these formulas see the papers of L. Toth.
a(n) = Sum_{d|n} psi(d), where psi is Dedekind's psi function A001615. - Peter Luschny, Sep 10 2012
a(n) = Sum_{d|n} 2^omega(d)*(n/d). - Peter Luschny, Sep 15 2012
Sum_{k=1..n} a(k) ~ (5/4) * n^2. - Amiram Eldar, Oct 19 2022
a(n) = Sum_{k=1..n} tau(gcd(n,k)^2). - Ridouane Oudra, Apr 10 2023
a(n) = Sum_{d divides n} J_2(d)/phi(d) = Sum_{1 <= i, j <= n} 1/phi(n/gcd(i,j,n)), where the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 23 2024

More terms and additional comments from Vladeta Jovovic, Jul 05 2001

A050488 a(n) = 3(2^n-1) - 2n.

Original entry on oeis.org

0, 1, 5, 15, 37, 83, 177, 367, 749, 1515, 3049, 6119, 12261, 24547, 49121, 98271, 196573, 393179, 786393, 1572823, 3145685, 6291411, 12582865, 25165775, 50331597, 100663243, 201326537, 402653127, 805306309, 1610612675, 3221225409, 6442450879, 12884901821, 25769803707

Offset: 0

James Sellers, Dec 26 1999

Number of words of length n+1 where first element is from {0,1,2}, other elements are from {0,1} and sequence does not decrease (for n=2 there are 3*2^2 sequences, but 000, 100, 110, 111, 200, 210, 211 decrease, so a(2) = 12-7 = 5).
Number of subgroups of C_(2^n) X C_(2^n) (see A060724).
Starting with 1 = row sums of triangle A054582. - Gary W. Adamson, Jun 23 2008
Starting with "1" equals the eigensequence of a triangle with integer squares (1, 4, 9, 16, ...) as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010
(1 + 2x + 2x^2 + 2x^3 + ...)*(1 + 3x + 7x^2 + 15x^3 + ...) = (1 + 5x + 15x^2 + 37x^3 + ...). - Gary W. Adamson, Mar 14 2012
The partial sums of A033484. - J. M. Bergot, Oct 03 2012
Binomial transform is 0, 1, 7, 33, ... (shifted A066810); inverse binomial transform is 0, 1, 3, 3, ... (3 repeated). - R. J. Mathar, Oct 05 2012
Define a triangle by T(n,0) = n*(n+1) + 1, T(n,n) = n + 1, and T(r,c) = T(r-1,c-1) + T(r-1,c) otherwise; then a(n+1) is the sum of the terms of row n. - J. M. Bergot, Mar 30 2013
Starting with "1" are also the antidiagonal sums of the array formed by partial sums of integer squares (1, 4, 9, 16, ...). - Luciano Ancora, Apr 24 2015
Sums of 2 adjacent terms in diagonal k=2 of Eulerian triangle A008292. I.e., T(n,2)+T(n-1,2) for n > 0. Also, 4th NW-SE diagonal of A126277. In other words, a(n) = A000295(n) + A000295(n+1). - Gregory Gerard Wojnar, Sep 30 2018

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Adam Buck, Jennifer Elder, Azia A. Figueroa, Pamela E. Harris, Kimberly Harry, and Anthony Simpson, Flattened Stirling Permutations, arXiv:2306.13034 [math.CO], 2023. See p. 14.
Tamas Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015) 73-94.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

A050487(2^m-1).
Equals (1/2) A051667.
Cf. A000079, A000225, A000295, A005408, A008292, A033484, A054852, A060724, A066810, A125165, A126277, A142963, A156919, A156920, A156925.

List([0..30],n->3*(2^n-1)-2*n); # Muniru A Asiru, Oct 26 2018

a050488 n = sum $ zipWith (*) a000079_list (reverse $ take n a005408_list)
-- Reinhard Zumkeller, Jul 24 2015

[3*(2^n-1) - 2*n: n in [0..30]]; // G. C. Greubel, Oct 23 2018

seq(coeff(series(x*(x+1)/((1-x)^2*(1-2*x)),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 26 2018

Table[3(2^n-1)-2n,{n,0,30}] (* or *) LinearRecurrence[{4,-5,2}, {0,1,5}, 40] (* Harvey P. Dale, Apr 09 2018 *)

a(n)=3*(2^n-1)-2*n \\ Charles R Greathouse IV, Sep 24 2015

for n in range(0, 30): print(3*(2**n-1) - 2*n, end=', ') # Stefano Spezia, Oct 27 2018

Row sums of A125165: (1, 5, 15, 37, ...). Binomial transform of [1, 4, 6, 6, 6, ...] = [1, 5, 15, 37, ...]. 4th diagonal from the right of A126777 = (1, 5, 15, ...). - Gary W. Adamson, Dec 23 2006
a(n) = 2*a(n-1) + (2n-1). - Gary W. Adamson, Sep 30 2007
From Johannes W. Meijer, Feb 20 2009: (Start)
a(n+1) = A156920(n+1,1).
a(n+1) = A156919(n+1,1)/2^n.
a(n+1) = A142963(n+2,1)/2.
a(n) = 4a(n-1) - 5a(n-2) + 2a(n-3) for n>2 with a(0) = 0, a(1) = 1, a(2) = 5.
G.f.: z*(1+z)/((1-z)^2*(1-2*z)).
(End)
a(n) = 2*n + 2*a(n-1) - 1 (with a(0)=0). - Vincenzo Librandi, Aug 06 2010
a(n+1) = Sum_{k=0..n} A000079(k) * A005408(n-k), convolution of the powers of 2 with the odd numbers. - Reinhard Zumkeller, Mar 08 2012
E.g.f.: exp(x)*(3*exp(x) - 2*x - 3). - Stefano Spezia, May 15 2023

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.

A064969 Number of cyclic 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, 8, 14, 36, 32, 112, 58, 148, 131, 256, 134, 504, 184, 464, 448, 596, 308, 1048, 382, 1152, 812, 1072, 554, 2072, 807, 1472, 1184, 2088, 872, 3584, 994, 2388, 1876, 2464, 1856, 4716, 1408, 3056, 2576, 4736, 1724, 6496, 1894, 4824, 4192, 4432, 2258, 8344

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 30 2001

Inverse Moebius transform of A160889. - Vladeta Jovovic, Nov 21 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110 and arXiv, arXiv:1103.5861 [math.NT], 2011.
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv:1203.6201 [math.GR], 2012.

Cf. A000010, A002117, A064803, A060648, A060724, A064767, A160889, A280184, A330595, A344219, A344302, A344303.

with(numtheory):
# define Jordan totient function J(r,n)
J(r,n) := add(d^r*mobius(n/d), d in divisors(n)):
seq(add(J(3,d)/phi(d), d in divisors(n)), n = 1..50); # Peter Bala, Jan 23 2024

a[n_] := Sum[EulerPhi[i] EulerPhi[j] (EulerPhi[k] / EulerPhi[LCM[i, j, k]]), {i, Divisors[n]}, {j, Divisors[n]}, {k, Divisors[n]}];
Array[a, 48] (* Jean-François Alcover, Dec 13 2018, after Vladeta Jovovic *)
f[p_, e_] := 1 + (p^2 + p + 1)*((p^(2*e) - 1)/(p^2 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, eulerphi(i)*eulerphi(j)*eulerphi(k)/eulerphi(lcm(lcm(i, j), k))))); \\ Michel Marcus, Dec 14 2018

a160889(n) = sumdiv(n, d, moebius(n/d)*d^3)/eulerphi(n);
a(n) = sumdiv(n, d, a160889(d)); \\ Seiichi Manyama, May 12 2021

a(n) = Sum_{i|n, j|n, k|n} phi(i)*phi(j)*phi(k)/phi(lcm(i, j, k)), where phi is Euler totient function (cf. A000010).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + (p^2 + p + 1)*((p^(2*e) - 1)/(p^2 - 1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^2 + 1/p^3) = A002117 * A330595 / 3 = 0.700772... . (End)
a(n) = Sum_{d divides n} J_3(d)/phi(d) = Sum_{1 <= i, j, k <= n} 1/phi(n/gcd(i,j,k,n)), where the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 23 2024

Formula and more terms from Vladeta Jovovic, Oct 30 2001

A344132 a(n) = Sum_{i|n, j|n, k|n} gcd(i,j,k).

Original entry on oeis.org

1, 9, 10, 37, 12, 90, 14, 111, 49, 108, 18, 370, 20, 126, 120, 283, 24, 441, 26, 444, 140, 162, 30, 1110, 79, 180, 184, 518, 36, 1080, 38, 657, 180, 216, 168, 1813, 44, 234, 200, 1332, 48, 1260, 50, 666, 588, 270, 54, 2830, 117, 711, 240, 740, 60, 1656, 216, 1554, 260, 324, 66, 4440, 68, 342, 686, 1441, 240, 1620, 74

Offset: 1

Seiichi Manyama, May 10 2021

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

Cf. A060724, A344133, A344134, A344135, A344138, A344139, A344140.

a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^3 &]; Array[a, 50] (* Amiram Eldar, May 10 2021 *)

a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, gcd([i, j, k]))));

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

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

a(n) = sum(k=1, n, numdiv(gcd(k, n))^3);

a(n) = Sum_{i|n, j|n, k|n} n/lcm(i,j,k).
a(n) = Sum_{d|n} phi(n/d) * tau(d)^3.
If p is prime, a(p) = 7 + p.
a(n) = Sum_{k=1..n} tau(gcd(k,n))^3.

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)]

A344140 a(n) = Sum_{x_1|n, x_2|n, ... , x_n|n} gcd(x_1,x_2, ... ,x_n).

Original entry on oeis.org

1, 5, 10, 99, 36, 4290, 134, 72613, 20713, 1053700, 2058, 2194638822, 8204, 268550150, 1073938440, 156969213515, 131088, 101697785139535, 524306, 3657271905119820, 4398063288332, 17592232181770, 8388630, 4727105990672866963914, 847422827191, 4503600499785740

Offset: 1

Seiichi Manyama, May 10 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..719

Cf. A000203, A060724, A344080, A344132, A344138, A344139.

a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^n &]; Array[a, 20] (* Amiram Eldar, May 10 2021 *)

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

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

a(n) = Sum_{x_1|n, x_2|n, ... , x_n|n} n/lcm(x_1,x_2, ... ,x_n).
a(n) = Sum_{d|n} phi(n/d) * tau(d)^n.
If p is prime, a(p) = 2^p - 1 + p.
a(n) = Sum_{k=1..n} tau(gcd(k,n))^n.

A064950 a(n) = Sum_{i|n, j|n} lcm(i,j).

Original entry on oeis.org

1, 7, 10, 27, 16, 70, 22, 83, 55, 112, 34, 270, 40, 154, 160, 227, 52, 385, 58, 432, 220, 238, 70, 830, 141, 280, 244, 594, 88, 1120, 94, 579, 340, 364, 352, 1485, 112, 406, 400, 1328, 124, 1540, 130, 918, 880, 490, 142, 2270, 267, 987, 520, 1080, 160, 1708

Offset: 1

Vladeta Jovovic, Oct 28 2001

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

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

a[n_]:= Sum[LCM[i,j], {i, Divisors[n]}, {j, Divisors[n]}];
Array[a,60] (* Jean-François Alcover, Jun 03 2019 *)
f[p_, e_] := (p^(e+2) - 3*p^(e+1) + p + 1 + 2*p^(e+2)*e - 2*p^(e+1)*e)/(p-1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Aug 28 2023 *)

for (n=1, 1000, d=divisors(n); a=sum(i=1, length(d), numdiv(d[i]^2)*d[i]); write("b064950.txt", n, " ", a)) \\ Harry J. Smith, Oct 01 2009

def A064950(n) :
    tau = sloane.A000005; D = divisors(n)
    return reduce(lambda x,y: x+y, [d*tau(d^2) for d in D])
[A064950(n) for n in (1..54)] # Peter Luschny, Sep 10 2012

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

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

A344139 a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n, x_5|n} gcd(x_1,x_2,x_3,x_4,x_5).

Original entry on oeis.org

1, 33, 34, 277, 36, 1122, 38, 1335, 313, 1188, 42, 9418, 44, 1254, 1224, 4771, 48, 10329, 50, 9972, 1292, 1386, 54, 45390, 391, 1452, 1720, 10526, 60, 40392, 62, 14193, 1428, 1584, 1368, 86701, 68, 1650, 1496, 48060, 72, 42636, 74, 11634, 11268, 1782, 78, 162214, 477, 12903, 1632, 12188, 84

Offset: 1

Seiichi Manyama, May 10 2021

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

Cf. A000203, A060724, A344132, A344138, A344140.

a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^5 &]; Array[a, 20] (* Amiram Eldar, May 10 2021 *)

a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, sumdiv(n, l, sumdiv(n, m, gcd([i, j, k, l, m]))))));

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

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

a(n) = sum(k=1, n, numdiv(gcd(k, n))^5);

a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n, x_5|n} n/lcm(x_1,x_2,x_3,x_4,x_5).
a(n) = Sum_{d|n} phi(n/d) * tau(d)^5.
If p is prime, a(p) = 31 + p.
a(n) = Sum_{k=1..n} tau(gcd(k,n))^5.

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

A060648 Number of cyclic subgroups of the group C_n X C_n (where C_n is the cyclic group of order n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

Extensions

A050488 a(n) = 3*(2^n-1) - 2*n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A064969 Number of cyclic subgroups of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A344132 a(n) = Sum_{i|n, j|n, k|n} gcd(i,j,k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A050488 a(n) = 3(2^n-1) - 2n.