A060648 -id:A060648 - cp's OEIS frontend

A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, 72, 38, 60, 56, 72, 42, 96, 44, 72, 72, 72, 48, 96, 56, 90, 72, 84, 54, 108, 72, 96, 80, 90, 60, 144, 62, 96, 96, 96, 84, 144, 68, 108, 96

Offset: 1

N. J. A. Sloane

Number of primitive sublattices of index n in generic 2-dimensional lattice; also index of Gamma_0(n) in SL_2(Z).
A generic 2-dimensional lattice L =  consists of all vectors of the form mV + nW, (m,n integers). A sublattice S =  has index |ad-bc| and is primitive if gcd(a,b,c,d) = 1. The generic lattice L has precisely a(2) = 3 sublattices of index 2, namely <2V,W>,  and  (which = ) and so on for other indices.
The sublattices of index n are in 1-to-1 correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} = sigma(n), which is A000203. A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * product_{p|n} (1+1/p), which is the present sequence.
SL_2(Z) = Gamma is the group of all 2 X 2 matrices [a b; c d] where a,b,c,d are integers with ad-bc = 1 and Gamma_0(N) is usually defined as the subgroup of this for which N|c. But conceptually Gamma is best thought of as the group of (positive) automorphisms of a lattice , its typical element taking V -> aV + bW, W -> cV + dW and then Gamma_0(N) can be defined as the subgroup consisting of the automorphisms that fix the sublattice  of index N. - J. H. Conway, May 05 2001
Dedekind proved that if n = k_i*j_i for i in I represents all the ways to write n as a product, and e_i=gcd(k_i,j_i), then a(n)= sum(k_i / (e_i * phi(e_i)), i in I ) [cf. Dickson, History of the Theory of Numbers, Vol. 1, p. 123].
Also a(n)= number of cyclic subgroups of order n in an Abelian group of order n^2 and type (1,1) (Fricke). - Len Smiley, Dec 04 2001
The polynomial degree of the classical modular equation of degree n relating j(z) and j(nz) is psi(n) (Fricke). - Michael Somos, Nov 10 2006; clarified by Katherine E. Stange, Mar 11 2022
The Mobius transform of this sequence is A063659. - Gary W. Adamson, May 23 2008
The inverse Mobius transform of this sequence is A060648. - Vladeta Jovovic, Apr 05 2009
The Dirichlet inverse of this sequence is A008836(n) * A048250(n). - Álvar Ibeas, Mar 18 2015
The Riemann Hypothesis is true if and only if a(n)/n - e^gamma*log(log(n)) < 0 for any n > 30. - Enrique Pérez Herrero, Jul 12 2011
The Riemann Hypothesis is also equivalent to another inequality, see the Sole and Planat link. - Thomas Ordowski, May 28 2017
An infinitary analog of this sequence is the sum of the infinitary divisors of n (see A049417). - Vladimir Shevelev, Apr 01 2014
Problem: are there composite numbers n such that n+1 divides psi(n)? - Thomas Ordowski, May 21 2017
The sum of divisors d of n such that n/d is squarefree. - Amiram Eldar, Jan 11 2019
Psi(n)/n is a new maximum for each primorial (A002110) [proof in link: Patrick Sole and Michel Planat, Proposition 1 page 2]. - Bernard Schott, May 21 2020
From Jianing Song, Nov 05 2022: (Start)
a(n) is the number of subgroups of C_n X C_n that are isomorphic to C_n, where C_n is the cyclic group of order n. Proof: the number of elements of order n in C_n X C_n is A007434(n) (they are the elements of the form (a,b) in C_n X C_n where gcd(a,b,n) = 1), and each subgroup isomorphic to C_n contains phi(n) generators, so the number of such subgroups is A007434(n)/phi(n) = a(n).
The total number of order-n subgroups of C_n X C_n is A000203(n). (End)

			Let L = <V,W> be a 2-dimensional lattice. The 6 primitive sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V+W,2W>, <2V,2W+V>. Compare A000203.
G.f. = x + 3*x^2 + 4*x^3 + 6*x^4 + 6*x^5 + 12*x^6 + 8*x^7 + 12*x^8 + 12*x^9 + ...

Tom Apostol, Intro. to Analyt. Number Theory, page 71, Problem 11, where this is called phi_1(n).
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 228.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 220.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 79.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (1).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..10000
O. Bordelles and B. Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq. 16 (2013) #13.6.3.
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, Representing and counting the subgroups of the group Z_m X Z_n, arXiv:1211.1797 [math.GR], 2012.
W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications 14:2 (2015), 73-88.
F. A. Lewis et al., Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
E. Pérez Herrero, Recycling Hardy & Wright, Average Order of Dedekind Psi Function, Psychedelic Geometry Blogspot.
Michel Planat, Riemann hypothesis from the Dedekind psi function, arXiv:1010.3239 [math.GM], 2010.
Patrick Sole and Michel Planat, Extreme values of the Dedekind Psi function, to appear in Journal of Combinatorics and Number Theory, arXiv:1011.1825 [math.NT], 2010-2011.
Eric Weisstein's World of Mathematics, Dedekind Function
Wikipedia, Dedekind psi function
Index entries for "core" sequences
Index entries for sequences related to sublattices

Other sequences that count lattices/sublattices: A000203 (with primitive condition removed), A003050 (hexagonal lattice instead), A003051, A054345, A160889, A160891.
Cf. A002110, A027748, A124010, A019269.
Cf. A301594.
Cf. A063659 (Möbius transform), A082020 (average order), A156303 (Euler transform), A173290 (partial sums), A175836 (partial products), A203444 (range).
Cf. A210523 (record values).
Algebraic combinations with other core sequences: A000082, A033196, A175732, A291784, A344695.
Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), this sequence (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).
Cf. A082695 (Dgf at s=3), A339925 (Dgf at s=4).

import Data.Ratio (numerator)
a001615 n = numerator (fromIntegral n * (product $
            map ((+ 1) . recip . fromIntegral) $ a027748_row n))
-- Reinhard Zumkeller, Jun 03 2013, Apr 12 2012

m:=75; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[MoebiusMu(k)^2*x^k/(1-x^k)^2: k in [1..2*m]]) )); // G. C. Greubel, Nov 23 2018

A001615 := proc(n) n*mul((1+1/i[1]),i=ifactors(n)[2]) end; # Mark van Hoeij, Apr 18 2012

Join[{1}, Table[n Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]), {n, 2, 100}]] (* T. D. Noe, Jun 11 2006 *)
Table[DirichletConvolve[j, MoebiusMu[j]^2, j, n], {n, 100}] (* Jan Mangaldan, Aug 22 2013 *)
a[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}]; (* Michael Somos, Jan 10 2015 *)
Table[n Product[1 + 1/p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 08 2021 *)
Table[n DivisorSum[n, MoebiusMu[#]^2/# &], {n, 20}] (* Eric W. Weisstein, Mar 09 2025 *)

{a(n) = if( n<1, 0, direuler( p=2, n, (1 + X) / (1 - p*X)) [n])};

{a(n) = if( n<1, 0, n * sumdiv( n, d, moebius(d)^2 / d))}; /* Michael Somos, Nov 10 2006 */

a(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]^f[i,2] + f[i,1]^(f[i,2]-1)) \\ Charles R Greathouse IV, Aug 22 2013

a(n) = n * sumdivmult(n, d, issquarefree(d)/d) \\ Charles R Greathouse IV, Sep 09 2014

from math import prod
from sympy import primefactors
def A001615(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist) # Chai Wah Wu, Jun 03 2021

def A001615(n) : return n*mul(1+1/p for p in prime_divisors(n))
[A001615(n) for n in (1..69)] # Peter Luschny, Jun 10 2012

Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(2*s). - Michael Somos, May 19 2000
Multiplicative with a(p^e) = (p+1)*p^(e-1). - David W. Wilson, Aug 01 2001
a(n) = A003557(n)*A048250(n) = n*A000203(A007947(n))/A007947(n). - Labos Elemer, Dec 04 2001
a(n) = n*Sum_{d|n} mu(d)^2/d, Dirichlet convolution of A008966 and A000027. - Benoit Cloitre, Apr 07 2002
a(n) = Sum_{d|n} mu(n/d)^2 * d. - Joerg Arndt, Jul 06 2011
From Enrique Pérez Herrero, Aug 22 2010: (Start)
a(n) = J_2(n)/J_1(n) = J_2(n)/phi(n) = A007434(n)/A000010(n), where J_k is the k-th Jordan Totient Function.
a(n) = (1/phi(n))*Sum_{d|n} mu(n/d)*d^(b-1), for b=3. (End)
a(n) = n / Sum_{d|n} mu(d)/a(d). - Enrique Pérez Herrero, Jun 06 2012
a(n^k)= n^(k-1) * a(n). - Enrique Pérez Herrero, Jan 05 2013
If n is squarefree, then a(n) = A049417(n) = A000203(n). - Vladimir Shevelev, Apr 01 2014
a(n) = Sum_{d^2 | n} mu(d) * A000203(n/d^2). - Álvar Ibeas, Dec 20 2014
The average order of a(n) is 15*n/Pi^2. - Enrique Pérez Herrero, Jan 14 2012. See Apostol. - N. J. A. Sloane, Sep 04 2017
G.f.: Sum_{k>=1} mu(k)^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Oct 25 2018
a(n) = Sum_{d|n} 2^omega(d) * phi(n/d), Dirichlet convolution of A034444 and A000010. - Daniel Suteu, Mar 09 2019
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} 2^omega(gcd(n,k)).
a(n) = Sum_{k=1..n} 2^omega(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = abs(A158523(n)) = A158523(n) * A008836(n). - Enrique Pérez Herrero, Nov 07 2022
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^2). - Ridouane Oudra, Mar 26 2025

More terms from Olivier Gérard, Aug 15 1997

A062011 a(n) = 2tau(n) = 2A000005(n).

Original entry on oeis.org

2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 12, 4, 8, 8, 10, 4, 12, 4, 12, 8, 8, 4, 16, 6, 8, 8, 12, 4, 16, 4, 12, 8, 8, 8, 18, 4, 8, 8, 16, 4, 16, 4, 12, 12, 8, 4, 20, 6, 12, 8, 12, 4, 16, 8, 16, 8, 8, 4, 24, 4, 8, 12, 14, 8, 16, 4, 12, 8, 16, 4, 24, 4, 8, 12, 12, 8, 16, 4, 20, 10, 8, 4, 24, 8, 8, 8, 16

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001

Old definition was "Number of cyclic subgroups of the group C_n X C_2 (where C_n is the cyclic group with n elements)."
More generally, the number of cyclic subgroups of the group C_n X C_m is Sum_{i|n, j|m} phi(i)*phi(j)/phi(lcm(i,j)), where phi=Euler totient function, cf. A000010. - Vladeta Jovovic, Jul 15 2001
Number of divisors of p*n, where p is any prime not dividing n. - Reinhard Zumkeller, May 17 2006
From Enrique Pérez Herrero, Jul 21 2011: (Start)
If p(x) is a polynomial with integer coefficients, and if r is an integer zero of p(x), then r is a divisor of the constant term c_0 of p(x). Under this theorem, p(x) can have a(c_0) possible integer roots.
a(n) is the number of integer divisors of n, while A000005(n) is the number of positive divisors. (End)
Number of solutions to the Diophantine equation i*j = n*i + j. - Robert G. Wilson v, Apr 10 2019
a(n) is also the number of times n appears in the triangle A333119, or equivalently, the number of positive integer solutions of the equation A333119(x, y) = n for y < x. - Stefano Spezia, Oct 05 2022

Harry J. Smith, Table of n, a(n) for n = 1..1000
Omar E. Pol, Illustration of initial terms
Index entries for sequences related to groups

Cf. A000005, A060648, A060710.
Cf. A087560, A119416.
Cf. A333119.

A062011[n_] := 2*DivisorSigma[0,n]; Array[A062011,50] (* Enrique Pérez Herrero, Jul 15 2011 *)

for (n=1, 1000, write("b062011.txt", n, " ", 2*numdiv(n))) \\ Harry J. Smith, Jul 29 2009

a(n) = A000005(A087560(n)) = A000005(A119416(n)). - Reinhard Zumkeller, May 17 2006

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(2/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 18 2018

More terms from Vladeta Jovovic, Jul 14 2001

Edited by N. J. A. Sloane, Sep 20 2018, replacing old definition (which was of course correct) with a simple formula.

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

Original entry on oeis.org

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

A062380 a(n) = Sum_{i|n,j|n} phi(i)*phi(j)/phi(gcd(i,j)), where phi is Euler totient function.

Original entry on oeis.org

1, 4, 7, 14, 13, 28, 19, 42, 37, 52, 31, 98, 37, 76, 91, 114, 49, 148, 55, 182, 133, 124, 67, 294, 113, 148, 163, 266, 85, 364, 91, 290, 217, 196, 247, 518, 109, 220, 259, 546, 121, 532, 127, 434, 481, 268, 139, 798, 229, 452, 343, 518, 157, 652, 403, 798, 385

Offset: 1

Vladeta Jovovic, Jul 07 2001

A176003 is a subsequence. - Peter Luschny, Sep 12 2012

			Let p be a prime then a(p) = phi(1)*tau(1)+phi(p)*tau(p^2) = 1+(p-1)*3 = 3*p-2. - _Peter Luschny_, Sep 12 2012

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

Cf. A000005, A000010, A060648, A062949.

with(numtheory):
a:= n->  add(phi(d)*tau(d^2), d=divisors(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 12 2012

a[n_] := DivisorSum[n, EulerPhi[#] DivisorSigma[0, #^2]&]; Array[a, 60] (* Jean-François Alcover, Dec 05 2015 *)
f[p_, e_] := ((2*e+1)*p^(e+1) - (2*e+3)*p^e + 2)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

a(n)=sumdiv(n,i,eulerphi(i)*sumdiv(n,j,eulerphi(j)/eulerphi(gcd(i,j)))) \\ Charles R Greathouse IV, Sep 12 2012

def A062380(n) :
    d = divisors(n); cp = cartesian_product([d, d])
    return reduce(lambda x,y: x+y, map(euler_phi, map(lcm, cp)))
[A062380(n) for n in (1..57)]  # Peter Luschny, Sep 10 2012

a(n) = Sum_{d|n} phi(d)*tau(d^2).
Multiplicative with a(p^e) = 1 + Sum_{k=1..e} (2k+1)(p^k-p^{k-1}) = ((2e+1)p^(e+1) - (2e+3)p^e+2)/(p-1). - Mitch Harris, May 24 2005
a(n) = Sum_{c|n,d|n} phi(lcm(c,d)). - Peter Luschny, Sep 10 2012
a(n) = Sum_{k=1..n} tau( (n/gcd(k,n))^2 ). - Seiichi Manyama, May 19 2024

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

A156302 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^2*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 5, 10, 30, 57, 152, 289, 676, 1304, 2809, 5335, 10961, 20487, 40329, 74476, 141914, 258094, 479638, 860025, 1563716, 2767982, 4940567, 8636563, 15173805, 26217392, 45416811, 77629455, 132800937, 224695510, 380079521, 637006921

Offset: 0

Paul D. Hanna, Feb 08 2009

nonn

Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

			G.f.: A(x) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 57*x^5 + 152*x^6 +...
log(A(x)) = x + 3^2*x^2/2 + 4^2*x^3/3 + 7^2*x^4/4 + 6^2*x^5/5 + 12^2*x^6/6 +...
Also log(A(x)) = (x + 3*x^2 + 4*x^3 + 7*x^4 +...+ sigma(k)*x^k +...)/1 +
(3*x^2 + 7*x^4 + 12*x^6 + 15*x^8 + 18*x^10 +...+ sigma(2*k)*x^(2*k) +...)/2 +
(4*x^3 + 12*x^6 + 13*x^9 + 28*x^12 + 24*x^15 +...+ sigma(3*k)*x^(3*k) +...)/3 +
(7*x^4 + 15*x^8 + 28*x^12 + 31*x^16 + 42*x^20 +...+ sigma(4*k)*x^(4*k) +...)/4 +
(6*x^5 + 18*x^10 + 24*x^15 + 42*x^20 + 31*x^25 +...+ sigma(5*k)*x^(5*k) +...)/5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000203 (sigma), A000041 (partitions), A072861, A178933, A205797, A382125.

nmax = 40; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[1, k]^2*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2024 *)

{a(n)=polcoeff(exp(sum(k=1,n,sigma(k)^2*x^k/k)+x*O(x^n)),n)}

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k)^2*a(n-k)))}

{a(n)=polcoeff(exp(sum(m=1,n+1,sum(k=1,n\m,sigma(m*k)*x^(m*k)/m)+x*O(x^n))),n)}

a(n) = (1/n)*Sum_{k=1..n} sigma(k)^2*a(n-k) for n>0, with a(0) = 1.
Euler transform of A060648. [From Vladeta Jovovic, Feb 14 2009]
It appears that G.f.: A(x)=prod(n=1,infinity, E(x^n)^(-A001615(n))) where E(x) = prod(n=1,infinity,1-x^n). [From Joerg Arndt, Dec 30 2010]
G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k) * x^(n*k) / n ). [From Paul D. Hanna, Jan 23 2012]
log(a(n)) ~ 3*(5*zeta(3))^(1/3) * n^(2/3) / 2. - Vaclav Kotesovec, Oct 29 2024

A175836 a(n) = Product_{i=1..n} psi(i) where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 3, 12, 72, 432, 5184, 41472, 497664, 5971968, 107495424, 1289945088, 30958682112, 433421549568, 10402117189632, 249650812551168, 5991619501228032, 107849151022104576, 3882569436795764736

Offset: 1

Enrique Pérez Herrero, Sep 18 2010

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = A060648(gcd(i,j)) for 1 <= i,j <= n, note that A060648 is the Inverse Möbius transform of A001615. - Enrique Pérez Herrero, Aug 12 2011

Charles R Greathouse IV, Table of n, a(n) for n = 1..423
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49
Eric Weisstein's World of Mathematics, Le Paige's Theorem
Index to divisibility sequences

Cf. A001615, A001088, A059381, A059382, A059383, A059384, A238498.

Cf. A239672.

a175836 n = a175836_list !! (n-1)
a175836_list = scanl1 (*) a001615_list
-- Reinhard Zumkeller, Mar 01 2014

A175836 := proc(n) option remember; local p; `if`(n<2,1, n*mul(1+1/p,p=factorset(n))*A175836(n-1)) end: # Peter Luschny, Feb 28 2014

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n];
DedekindPsi[n_]:=JordanTotient[n,2]/EulerPhi[n];
A175836[n_]:=Times@@DedekindPsi/@Range[n]

a=direuler(p=2, 100, (1+X)/(1-p*X));for(i=2,#a,a[i]*=a[i-1]);a
\\ Charles R Greathouse IV, Jul 29 2011

a(n) = A059381(n)/A001088(n).

A280184 Number of cyclic subgroups of the group C_n x 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, 41, 136, 157, 656, 401, 1096, 1121, 2512, 1465, 5576, 2381, 6416, 6437, 8776, 5221, 17936, 7241, 21352, 16441, 23440, 12721, 44936, 19657, 38096, 30281, 54536, 25261, 102992, 30785, 70216, 60065, 83536, 62957, 152456, 52061, 115856, 97621, 172072, 70645, 263056, 81401, 199240, 175997, 203536, 106081, 359816, 137601, 314512

Offset: 1

Laszlo Toth, Dec 28 2016

Inverse Moebius transform of A160891. - Seiichi Manyama, May 12 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110.
László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Cf. A000010, A013662, A060648, A064969, A160891, A280162, 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(4,d)/phi(d), d in divisors(n)), n = 1..50); # Peter Bala, Jan 23 2024

a[n_] := With[{dd = Divisors[n]}, Sum[Times @@ EulerPhi @ {x, y, z, t} / EulerPhi[LCM[x, y, z, t]], {x, dd}, {y, dd}, {z, dd}, {t, dd}]];
Array[a, 50] (* Jean-François Alcover, Sep 28 2018 *)
f[p_, e_] := 1 + (p^3 + p^2 + p + 1)*((p^(3*e) - 1)/(p^3 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, x, sumdiv(n, y, sumdiv(n, z, sumdiv(n, t, eulerphi(x)*eulerphi(y)*eulerphi(z)*eulerphi(t)/eulerphi(lcm([x, y, z, t])))))); \\ Michel Marcus, Feb 26 2018

a160891(n) = sumdiv(n, d, moebius(n/d)*d^4)/eulerphi(n);
a(n) = sumdiv(n, d, a160891(d)); \\ Seiichi Manyama, May 12 2021

a(n) = Sum_{a|n, b|n, c|n, d|n} phi(a)*phi(b)*phi(c)*phi(d)/phi(lcm(a, b, c, d)), where phi is Euler totient function (cf. A000010).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + (p^3 + p^2 + p + 1)*((p^(3*e) - 1)/(p^3 - 1)).
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)/4) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4) = 0.5010902655... . (End)
a(n) = Sum_{d divides n} J_4(d)/phi(d) = Sum_{1 <= i, j, k, l <= n} 1/phi(n/gcd(i,j,k,l,n)), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 23 2024

cp's OEIS Frontend

A306775 Partial sums of A060648: sum of the inverse Moebius transform of the Dedekind psi function from 1 to n.

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A307227 Numbers k such that A060648(k) is divisible by k.

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A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).

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A062011 a(n) = 2*tau(n) = 2*A000005(n).

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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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A062380 a(n) = Sum_{i|n,j|n} phi(i)*phi(j)/phi(gcd(i,j)), where phi is Euler totient function.

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A062011 a(n) = 2tau(n) = 2A000005(n).