%I A001615 M2315 N0915
%S A001615 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,
%T A001615 42,36,48,30,72,32,48,48,54,48,72,38,60,56,72,42,96,44,72,72,72,48,96,
%U A001615 56,90,72,84,54,108,72,96,80,90,60,144,62,96,96,96,84,144,68,108,96
%N A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).
%C A001615 Number of primitive sublattices of index n in generic 2-dimensional lattice; 
               also index of GAMMA_0(n) in SL_2(Z).
%C A001615 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.
%C A001615 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.
%C 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
%C A001615 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].
%C A001615 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
%C A001615 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
%C A001615 Mobius transform of A001615 = A063659. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 23 2008
%D A001615 D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989, p. 228.
%D A001615 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.
%D A001615 R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 
               1922, Vol. 2, see p. 220.
%D A001615 F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 
               9, Nov. 1942, pp. 618-619.
%D A001615 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, 
               p. 79.
%D A001615 G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, 
               Princeton, 1971, see p. 25, Eq. (1).
%D A001615 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001615 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001615 T. D. Noe and N. J. A. Sloane, <a href="b001615.txt">Table of n, a(n) 
               for n = 1..10000</a>
%H A001615 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DedekindFunction.html">Link to a section of The World of Mathematics.</
               a>
%H A001615 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A001615 <a href="Sindx_Su.html#sublatts">Index entries for sequences related 
               to sublattices</a>
%F A001615 Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(2*s) - Michael Somos, May 19, 
               2000
%F A001615 Multiplicative with a(p^e) = (p+1)*p^(e-1). - David W. Wilson (davidwwilson(AT)comcast.net), 
               Aug 01, 2001.
%F A001615 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
%F A001615 a(n) = n*sum(d|n, mu(d)^2/d) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 07 2002
%e A001615 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.
%p A001615 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
%p A001615 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
%p A001615 Join[{1}, Table[n Times@@(1+1/Transpose[FactorInteger[n]][[1]]), {n,2,
               100}]] - T. D. Noe (noe(AT)sspectra.com), Jun 11 2006
%o A001615 (PARI) a(n)=if(n<1,0,direuler(p=2,n,(1+X)/(1-p*X))[n])
%o A001615 (PARI) {a(n)=if(n<1, 0, n*sumdiv(n, d, moebius(d)^2/d))} /* Michael Somos 
               Nov 10 2006 */
%Y A001615 Cf. A003051, A003050, A054345, A000082, A033196, A000203.
%Y A001615 Cf. A063659.
%Y A001615 Sequence in context: A063649 A053158 A158523 this_sequence A133689 A135510 
               A065967
%Y A001615 Adjacent sequences: A001612 A001613 A001614 this_sequence A001616 A001617 
               A001618
%K A001615 nonn,easy,core,nice,mult
%O A001615 1,2
%A A001615 N. J. A. Sloane (njas(AT)research.att.com).
%E A001615 More terms and Mathematica program Aug 15 1997 (Olivier Gerard).