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A001615
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Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).
(Formerly M2315 N0915)
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54
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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
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OFFSET
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1,2
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COMMENTS
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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
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]
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 average order of a(n) is 15n/Pi^2. - Enrique Pérez Herrero, Jan 14 2012
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REFERENCES
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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.
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, 2012. - From N. J. A. Sloane, Jan 02 2013
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).
Patrick Sole and Michel Planat, Extreme values of the Dedekind Psi function, to appear in Journal of Combinatorics and Number Theory, arXiv:1011.1825v2.
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LINKS
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T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..10000
N. A. Carella, A Dedekind Psi Function Inequality, arXiv:1112.0208v2
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:1010.3239v2
Eric Weisstein's World of Mathematics, Dedekind Function
Wikipedia, Dedekind psi function
Index entries for "core" sequences
Index entries for sequences related to sublattices
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FORMULA
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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) = A000203(A007947(n))/A007947(n). - Labos E. (labos(AT)ana.sote.hu), 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]
Contribution 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) = sum(d|n,mu(n/d)*d^(b-1)/phi(n)), 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
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EXAMPLE
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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.
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 + ...
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MAPLE
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A001615 := proc(n) n*mul((1+1/i[1]), i=ifactors(n)[2]) end; # Mark van Hoeij, Apr 18 2012
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MATHEMATICA
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Join[{1}, Table[n Times@@(1+1/Transpose[FactorInteger[n]][[1]]), {n, 2, 100}]] (* T. D. Noe, Jun 11 2006 *)
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PROG
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(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 */
(Haskell)
a001615 n = product $ zipWith (*) (map (+ 1 ) $ a027748_row n)
(zipWith (^) (a027748_row n) (map (subtract 1) $ a124010_row n))
-- Reinhard Zumkeller, 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
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CROSSREFS
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Cf. A003051, A003050, A054345, A000082, A033196, A000203, A063659, A160889, A160891, A173290, A082020, A027748, A124010, A019269.
Sequence in context: A206924 A185443 A158523 * A133689 A220345 A135510
Adjacent sequences: A001612 A001613 A001614 * A001616 A001617 A001618
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KEYWORD
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nonn,easy,core,nice,mult
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms and Mathematica program Aug 15 1997 (Olivier Gerard).
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STATUS
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approved
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