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. 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 one-one 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 A001615.

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 represent all 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 (smiley(AT)math.uaa.alaska.edu), Dec 04 2001

The polynomial degree of the classical modular equation of degree n relating j(z) and j(nz) is denoted by psi(n) by Fricke. - Michael Somos Nov 10 2006

Mobius transform of A001615 = A063659. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23 2008

D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989, p. 228.

Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations. J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 220.

F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.

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

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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 (davidwwilson(AT)comcast.net), Aug 01, 2001.

a[n] = n*A048250(n)/A007947(n) = A000203[A007947(n)]/A007947(n); or a(n) = nProduct[1+(1/p)], p divides n; Dedekind-function. - Labos E. (labos(AT)ana.sote.hu), Dec 04 2001

a(n) = n*sum(d|n, mu(d)^2/d) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 07 2002

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.

with(numtheory): A001615 := proc(n) local i, j; j := n; for i in divisors(n) do if isprime(i) then j := j*(1+1/i); fi; od; j; end; # version 1

with(numtheory): A001615 := proc(n) local i, t1, t2; t1 := ifactors(n)[2]; t2 := n*mul((1+1/t1[i][1]), i=1..nops(t1)); end; # version 2

Join[{1}, Table[n Times@@(1+1/Transpose[FactorInteger[n]][[1]]), {n, 2, 100}]] - T. D. Noe (noe(AT)sspectra.com), Jun 11 2006

(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 */

Cf. A003051, A003050, A054345, A000082, A033196, A000203.

Cf. A063659.

Sequence in context: A063649 A053158 A158523 this_sequence A133689 A135510 A065967

Adjacent sequences: A001612 A001613 A001614 this_sequence A001616 A001617 A001618

nonn,easy,core,nice,mult

N. J. A. Sloane (njas(AT)research.att.com).

More terms and Mathematica program Aug 15 1997 (Olivier Gerard).