1,2

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 = <V,W> consists of all vectors of the form mV + nW, (m,n integers). A sublattice S = <aV+bW, cV+dW> 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>, <V,2W> and <V+W,2V> (which = <V+W,2W>) 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 <V,W>, 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 <NV,W> 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)

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

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

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 + ...

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_] := If[ n < 1, 0, 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 *)

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

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

(PARI) 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

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

(Haskell)

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

(Sage) 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

(Magma) m:=75; R<x>:=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

(Python 3.8+)

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

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).

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).

Sequence in context: A332619 A322319 A158523 * A133689 A285895 A220345

Adjacent sequences: A001612 A001613 A001614 * A001616 A001617 A001618

nonn,easy,core,nice,mult

N. J. A. Sloane

More terms from Olivier Gérard, Aug 15 1997

approved