A000082 -id:A000082 - cp's OEIS frontend

A140697 Mobius transform of A000082.

1, 5, 11, 18, 29, 55, 55, 72, 96, 145, 131, 198, 181, 275, 319, 288, 305, 480, 379, 522, 605, 655, 551, 792, 720, 905, 864, 990, 869, 1595, 991, 1152, 1441, 1525, 1595, 1728, 1405, 1895, 1991, 2088, 1721, 3025, 1891, 2358, 2784, 2755, 2255, 3168, 2688, 3600

Offset: 1

Gary W. Adamson, May 23 2008

Dirichlet convolution of the sequence of (absolute values of A055615) and A007434. - R. J. Mathar, Feb 27 2011

			a(4) = 18 = (0, -1, 0, 1) dot (1, 6, 12, 24), where (0, -1 0, 1) = row 4 of A054525 and A000082 = (1, 6, 12, 24, 30, 72,...).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000082, A002117, A007434, A054525, A055615.

with (numtheory): a:= n-> add (k^2* mul(1+1/p, p=factorset(k)) *mobius (n/k), k=divisors(n)): seq (a(n), n=1..60); # Alois P. Heinz, Aug 28 2008

a[n_] := Sum[ k^2*Product[ 1+1/p, {p, FactorInteger[k][[All, 1]]}]* MoebiusMu[n/k], {k, Divisors[n]}] - MoebiusMu[n]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 03 2012, after Alois P. Heinz *)
f[p_, e_] := (p - 1)*(p + 1)^2*p^(2*e - 3); f[p_, 1] := p*(p + 1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 28 2023 *)

Dirichlet g.f.: zeta(s-1)*zeta(s-2)/(zeta(2s-2)*zeta(s)). - R. J. Mathar, Feb 27 2011
Sum_{k=1..n} a(k) ~ 5*n^3 / (Pi^2 * zeta(3)). - Vaclav Kotesovec, Jan 11 2019
Multiplicative with a(p) = p*(p+1) - 1, and a(p^e) = (p-1)*(p+1)^2*p^(2*e-3) for e >= 2. - Amiram Eldar, Oct 28 2023

Definition corrected by N. J. A. Sloane, Jul 28 2008

More terms from Alois P. Heinz, Aug 28 2008

A377154 Expansion of e.g.f. exp(Sum_{k>=1} A000082(k)*x^k/k).

Original entry on oeis.org

1, 1, 7, 43, 385, 3721, 47911, 612067, 9559873, 157478545, 2910837511, 56866891291, 1224263236417, 27618866777113, 673173639519655, 17237263465417171, 469017851840595841, 13367670808113197857, 401964392506370969863, 12604372518766870306315, 414278024498330114803201

Offset: 0

Vaclav Kotesovec, Oct 31 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..428

Cf. A000082, A001615, A308462.

nmax = 25; CoefficientList[Series[Exp[Sum[k * Product[1 + 1/p, {p, Select[Divisors[k], PrimeQ]}] * x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ n! * 5^(1/6) * exp(-1/12 - 1/(20*Pi^2) - 3^(2/3)*n^(1/3) / (10^(1/3)*Pi^(4/3)) + 3^(4/3)*5^(1/3)*n^(2/3) / (2*Pi)^(2/3)) / (6^(1/3) * Pi^(5/6) * n^(2/3)).
a(n) ~ 10^(1/6) * exp(-1/12 - 1/(20*Pi^2) - 3^(2/3)*n^(1/3) / (10^(1/3)*Pi^(4/3)) + 3^(4/3)*5^(1/3)*n^(2/3) / (2*Pi)^(2/3)) * n^(n - 1/6) / (3^(1/3) * Pi^(1/3) * exp(n)).
E.g.f.: exp(Sum_{k>=1} A001615(k)*x^k).

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

Original entry on oeis.org

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

A065764 Sum of divisors of square numbers.

Original entry on oeis.org

1, 7, 13, 31, 31, 91, 57, 127, 121, 217, 133, 403, 183, 399, 403, 511, 307, 847, 381, 961, 741, 931, 553, 1651, 781, 1281, 1093, 1767, 871, 2821, 993, 2047, 1729, 2149, 1767, 3751, 1407, 2667, 2379, 3937, 1723, 5187, 1893, 4123, 3751, 3871, 2257, 6643

Offset: 1

Labos Elemer, Nov 19 2001

Unlike A065765, the sums of divisors of squares give remainders r=1,3,5 modulo 6: sigma(4)==1, sigma(49)==3, sigma(2401)==5 (mod 6). See also A097022.
a(n) is also the number of ordered pairs of positive integers whose LCM is n, (see LeVeque). - Enrique Pérez Herrero, Aug 26 2013
Main diagonal of A319526. - Omar E. Pol, Sep 25 2018
Subsequence of primes is A023195 \ {3}; also, 31 is the only known prime to be twice in the data because 31 = sigma(16) = sigma(25) (see A119598 and Goormaghtigh conjecture link). - Bernard Schott, Jan 17 2021

W. J. LeVeque, Fundamentals of Number Theory, pp. 125 Problem 4, Dover NY 1996.

T. D. Noe, Table of n, a(n) for n=1..10000
Wikipedia, Goormaghtigh conjecture.

Cf. A000203, A000290, A028982, A319526.

a:=List([1..50],n->Sigma(n^2));; Print(a); # Muniru A Asiru, Jan 01 2019

[SumOfDivisors(n^2): n in [1..48]]; // Bruno Berselli, Apr 12 2011

with(numtheory): [sigma(n^2)$n=1..50]; # Muniru A Asiru, Jan 01 2019

Table[Plus@@Divisors[n^2], {n, 48}] (* Alonso del Arte, Feb 24 2012 *)
f[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 10 2020 *)

numlib::sigma(n^2)$ n=1..81 // Zerinvary Lajos, May 13 2008

a(n) = sigma(n^2); \\ Harry J. Smith, Oct 30 2009

from math import prod
from sympy import factorint
def A065764(n): return prod((p**((e<<1)+1)-1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Oct 25 2023

[sigma(n^2,1)for n in range(1,49)] # Zerinvary Lajos, Jun 13 2009

a(n) = sigma(n^2) = A000203(A000290(n)).
Multiplicative with a(p^e) = (p^(2*e+1)-1)/(p-1). - Vladeta Jovovic, Dec 01 2001
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)/zeta(2*s-2), inverse Mobius transform of A000082. - R. J. Mathar, Mar 06 2011
Dirichlet convolution of A001157 by the absolute terms of A055615. Also the Dirichlet convolution of A048250 by A000290. - R. J. Mathar, Apr 12 2011
a(n) = Sum_{d|n} d*Psi(d), where Psi is A001615. - Enrique Pérez Herrero, Feb 25 2012
a(n) >= (n+1) * sigma(n) - n, where sigma is A000203, equality holds if n is in A000961. - Enrique Pérez Herrero, Apr 21 2012
Sum_{k=1..n} a(k) ~ 5*Zeta(3) * n^3 / Pi^2. - Vaclav Kotesovec, Jan 30 2019
Sum_{k>=1} 1/a(k) = 1.3947708738535614499846243600124612760835313454790187655653356563282177118... - Vaclav Kotesovec, Sep 20 2020

A159634 Coefficient for dimensions of spaces of modular & cusp forms of weight k/2, level 4*n and trivial character, where k>=5 is odd.

Original entry on oeis.org

1, 2, 4, 4, 6, 8, 8, 8, 12, 12, 12, 16, 14, 16, 24, 16, 18, 24, 20, 24, 32, 24, 24, 32, 30, 28, 36, 32, 30, 48, 32, 32, 48, 36, 48, 48, 38, 40, 56, 48, 42, 64, 44, 48, 72, 48, 48, 64, 56, 60, 72, 56, 54, 72, 72, 64, 80, 60, 60, 96, 62, 64, 96, 64, 84, 96, 68, 72, 96, 96

Offset: 1

Steven Finch, Apr 17 2009

Denote dim{M_k(Gamma_0(N))} by m(k,N) and dim{S_k(Gamma_0(N))} by s(k,N).
We have
m(7/2,N)+s(5/2,N) = m(5/2,N)+s(7/2,N) =
(m(11/2,N)+s(9/2,N))/2 = (m(9/2,N)+s(11/2,N))/2 =
(m(15/2,N)+s(13/2,N))/3 = (m(13/2,N)+s(15/2,N))/3 = ...
(m((4j+3)/2,N)+s((4j+1)/2,N))/j = (m((4j+1)/2,N)+s((4j+3)/2,N))/j = ...
where N is any positive multiple of 4 and j>=1.
Multiplicative because A001615 is multiplicative and a(1) = A001615(2)/3 = 1. - Andrew Howroyd, Aug 08 2018

Ken Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and q-series. American Mathematical Society, 2004, (p. 16, theorem 1.56).

Peter Luschny, Table of n, a(n) for n = 1..1000
H. Cohen and J. Oesterle, Dimensions des espaces de formes modulaires, Modular Functions of One Variable. VI, Proc. 1976 Bonn conf., Lect. Notes in Math. 627, Springer-Verlag, 1977, pp. 69-78; Scanned copy.
S. R. Finch, Primitive Cusp Forms, April 27, 2009. [Cached copy, with permission of the author]
Peter Humphries, Answer to: "A conjecture related to the Cohen-Oesterlé dimension formula", MathOverflow, 2014.
Jon Maiga, Computer-generated formulas for A159634, Sequence Machine.
Wikipedia, Cusp Form.

Cf. A159635, A159636. - Steven Finch, Apr 22 2009

Cf. A000010, A000082, A001221, A001615, A008683, A059956, A080512, A159630.

[[4*n,(Dimension(HalfIntegralWeightForms(4*n,7/2))+ Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,5/2))))/2] : n in [1..70]]; [[4*n,(Dimension(HalfIntegralWeightForms(4*n,5/2))+ Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,7/2))))/2] : n in [1..70]];

(* per Enrique Pérez Herrero's conjecture proved by P. Humphries, see link *)
dedekindPsi[n_Integer]:=n Apply[Times,1+1/Map[First,FactorInteger[n]]];
1/3 dedekindPsi /@ (2 Range[70]) (* Wouter Meeussen, Apr 06 2014 *)

a(n) = 2*n*sumdiv( 2*n, d, moebius(d)^2 / d)/3; \\ Andrew Howroyd, Aug 08 2018

a(n) = A159636(n) + A159630(n). - Enrique Pérez Herrero, Apr 15 2014
a(n) = A001615(2*n)/3. - Enrique Pérez Herrero, Jan 31 2014
From Peter Bala, Mar 19 2019: (Start)
a(n)= n*Product_{p|n, p odd prime} (1 + 1/p).
a(n) = Sum_{d|n, d odd} mu(d)^2*n/d, where mu(n) = A008683(n) is the Möbius function.
If n = m*2^k , where 2^k is the largest power of 2 dividing n, then
a(n) = (2^k)*a(m) = 2^k * Sum_{d^2|m} mu(d)*sigma(m/d^2), where sigma(n) = A000203(n) is the sum of the divisors of n, and also
a(n) = 2^k * Sum_{d|m} 2^omega(d)*phi(m/d), where omega(n) = A001221(n) is the number of different primes dividing n and phi(n) = A000010 is the Euler totient function.
O.g.f.: Sum_{n >= 1} mu(2*n-1)^2*x^(2^n-1)/(1 - x^(2*n-1))^2. (End)
a(n) = A000082(n)/A080512(n). [obvious by prime products, discovered by Sequence Machine]. - R. J. Mathar, Jun 24 2021
From Amiram Eldar, Nov 17 2022: (Start)
Multiplicative with a(2^e) = 2^e, and a(p^e) = (p+1)*p^(e-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 6/Pi^2 = 0.607927... (A059956). (End)

A181797 a(n) = n multiplied by the sum of its squarefree divisors (A048250(n)).

Original entry on oeis.org

1, 6, 12, 12, 30, 72, 56, 24, 36, 180, 132, 144, 182, 336, 360, 48, 306, 216, 380, 360, 672, 792, 552, 288, 150, 1092, 108, 672, 870, 2160, 992, 96, 1584, 1836, 1680, 432, 1406, 2280, 2184, 720, 1722, 4032, 1892, 1584, 1080, 3312, 2256, 576, 392, 900

Offset: 1

Matthew Vandermast, Dec 05 2010

Sum of reciprocals converges to Pi^2/6. The natural density of positive integers m such that A003557(m) = n equals 6/(a(n)*Pi^2).
If m is coprime to 6, a(3m) = a(4m).
Apparently the absolute values of the Dirichlet inverse of A000082. - R. J. Mathar, Mar 14 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Natural density

Cf. A181798, A181799.

A181797 := proc(n) local f; f := ifactors(n)[2] ;  mul( op(1,d)^op(2,d)*( op(1,d)+1),d=f) ; end proc: # R. J. Mathar, Dec 05 2010

Table[n*Sum[d*MoebiusMu[d]^2, {d, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, Feb 02 2019 *)

PARI
```
a(n)=n*sumdiv(n,d,d*moebius(d)^2)
```

A181797 = lambda n: n * sum(d for d in divisors(n) if is_squarefree(d)) # D. S. McNeil, Dec 05 2010

a(n) = n*A048250(n). Multiplicative with a(p^e) = (p+1)*p^e.
Dirichlet g.f. zeta(s-1)*zeta(s-2)/zeta(2*s-4). - R. J. Mathar, Mar 14 2011
G.f.: x*f'(x), where f(x) = Sum_{k>=1} mu(k)^2*k*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 10 2017
Sum_{k=1..n} a(k) ~ n^3 / 3. - Vaclav Kotesovec, Feb 02 2019
Sum_{k>=1} 1/a(k) = Pi^2/6. - Vaclav Kotesovec, Sep 19 2020

A033196 a(n) = n^3 * Product_{p|n, p prime} (1 + 1/p).

Original entry on oeis.org

1, 12, 36, 96, 150, 432, 392, 768, 972, 1800, 1452, 3456, 2366, 4704, 5400, 6144, 5202, 11664, 7220, 14400, 14112, 17424, 12696, 27648, 18750, 28392, 26244, 37632, 25230, 64800, 30752, 49152, 52272, 62424, 58800, 93312, 52022, 86640

Offset: 1

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000082, A001615.

a[n_] := n*DivisorSum[n, MoebiusMu[n/#] DivisorSigma[1, #^2]&]; Array[a, 40] (* Jean-François Alcover, Dec 02 2015 *)

a(n)=direuler(p=2,n,(1+p^2*X)/(1-p^3*X))[n]

a(n)=sumdiv(n,d,moebius(d)*sigma(n^3/d^2)) \\ Benoit Cloitre, Feb 16 2008

Dirichlet g.f.: zeta(s-2)*zeta(s-3)/zeta(2*s-4).
a(n) = n^2 * A001615(n) = n * A000082(n).
Multiplicative with a(p^e) = p^e*p^(2*e-1)*(p+1). - Vladeta Jovovic, Nov 16 2001
a(n) = Sum_{d|n} mu(d)*sigma(n^3/d^2). - Benoit Cloitre, Feb 16 2008
a(n) = A001615(n^3) = A001615(n^k)/n^(k-3), with k>2. - Enrique Pérez Herrero, Mar 06 2012
Sum_{k=1..n} a(k) ~ 15*n^4 / (4*Pi^2). - Vaclav Kotesovec, Feb 01 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p/((p+1)*(p^3-1))) = 1.1392293101137663761606045655621290749920977339371831842000361508083066155... - Vaclav Kotesovec, Sep 20 2020

Additional comments from Michael Somos, May 19 2000

A335762 Decimal expansion of Product_{p prime} (1 + p/((p-1)*(p+1)^2)).

Original entry on oeis.org

1, 4, 5, 0, 0, 3, 2, 1, 4, 5, 3, 6, 2, 1, 2, 0, 8, 3, 1, 6, 0, 8, 3, 9, 5, 8, 8, 7, 1, 8, 9, 2, 2, 3, 4, 2, 2, 3, 2, 5, 0, 6, 2, 1, 1, 7, 4, 4, 7, 1, 6, 7, 1, 4, 4, 6, 5, 2, 4, 3, 8, 8, 3, 6, 7, 0, 9, 4, 1, 6, 3, 3, 7, 2, 9, 3, 8, 0, 8, 3, 0, 7, 6, 8, 1, 3, 5, 8, 7, 0, 3, 6, 5, 5, 6, 3, 9, 1, 4, 6, 5, 5, 8, 5, 5

Offset: 1

Amiram Eldar, Jun 21 2020

The asymptotic mean of A367987. - Amiram Eldar, Dec 23 2023

			1.450032145362120831608395887189223422325062117447167...

Cf. A000082, A001615, A013661, A065465, A065484, A367987.

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{-1, 1, 2, 0, -1}, {0, 2, -3, 6, -5}, m]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n])/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

prodeulerrat(1 + p/((p-1)*(p+1)^2)) \\ Amiram Eldar, Mar 18 2021

Equals Sum_{k>=1} 1/A000082(k) = Sum_{k>=1} 1/(k * A001615(k)).

Equals A013661 * A065465. - Amiram Eldar, Dec 23 2023

More digits from Vaclav Kotesovec, Sep 19 2020

A076566 Greatest prime divisor of 3n+3 (sum of three successive integers).

Original entry on oeis.org

3, 3, 3, 5, 3, 7, 3, 3, 5, 11, 3, 13, 7, 5, 3, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 3, 11, 17, 7, 3, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 5, 17, 13, 53, 3, 11, 7, 19, 29, 59, 5, 61, 31, 7, 3, 13, 11, 67, 17, 23, 7, 71, 3, 73, 37, 5, 19, 11, 13, 79, 5, 3, 41, 83, 7

Offset: 1

Zak Seidov, Oct 19 2002

a(n) = A006530(A000082(n+1)). - Reinhard Zumkeller, Oct 03 2012

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

Cf. A008585.

a076566 = a006530 . (* 3) . (+ 1)  -- Reinhard Zumkeller, Oct 03 2012

FactorInteger[Total[#]][[-1,1]]&/@Partition[Range[100],3,1]  (* Harvey P. Dale, Apr 03 2011 *)

A070732 Size of largest conjugacy class in the group GL(2,Z_n).

Original entry on oeis.org

1, 3, 12, 12, 30, 36, 56, 48, 108, 90, 132, 144, 182, 168, 360, 192, 306, 324, 380, 360, 672, 396, 552, 576, 750, 546, 972, 672, 870, 1080, 992, 768, 1584, 918, 1680, 1296, 1406, 1140, 2184, 1440, 1722, 2016, 1892, 1584, 3240, 1656, 2256, 2304, 2744, 2250

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A062354.
Cf. A000010, A000056.
Cf. A000082, A099892, A098198, A330523.

f[n_] := Block[{a = 1, b = FactorInteger[n]}, While[ Length[b] > 0, a = a*(b[[1, 1]] + 1)*b[[1, 1]]^(2b[[1, 2]] - If[ OddQ[ b[[1, 1]]], 1, 2]); b = Drop[b, 1]]; a]; Table[ f[n], {n, 1, 55}]
Table[n*Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}]/EulerPhi[2*n], {n, 1, 100}] (* Vaclav Kotesovec, Feb 01 2019 *)
f[p_, e_] := (p + 1)*p^(2*e - If[p == 2, 2, 1]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]+1)*f[i,1]^(2*f[i,2] - if(f[i,1]==2,2,1)));} \\ Amiram Eldar, Nov 05 2022

Multiplicative with a(p^e) = (p+1)*p^(2e - k), k = 1 if p is odd, k = 2 if p is 2.
a(n) = A000056(n)/A000010(2*n). - Vladeta Jovovic, Dec 22 2003
From R. J. Mathar, Apr 14 2011: (Start)
Dirichlet g.f.: (2^s-1)*zeta(s-1)*zeta(s-2)/((2^s+2)*zeta(2s-2)).
Dirichlet convolution of A000082 with a signed variant of A099892. (End)
Sum_{k=1..n} a(k) ~ 7*n^3 / (2*Pi^2). - Vaclav Kotesovec, Feb 01 2019
Sum_{n>=1} 1/a(n) = (13/11) * zeta(2)^2 * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = (13/11) * A098198 * A330523 = 1.7136743536... . - Amiram Eldar, Nov 05 2022

Edited by Robert G. Wilson v, May 20 2002

cp's OEIS Frontend

A140697 Mobius transform of A000082.

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A377154 Expansion of e.g.f. exp(Sum_{k>=1} A000082(k)*x^k/k).

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

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A065764 Sum of divisors of square numbers.

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A159634 Coefficient for dimensions of spaces of modular & cusp forms of weight k/2, level 4*n and trivial character, where k>=5 is odd.

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A181797 a(n) = n multiplied by the sum of its squarefree divisors (A048250(n)).

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