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A007434
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Jordan function J_2(n) (a generalization of phi(n)).
(Formerly M2717)
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33
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1, 3, 8, 12, 24, 24, 48, 48, 72, 72, 120, 96, 168, 144, 192, 192, 288, 216, 360, 288, 384, 360, 528, 384, 600, 504, 648, 576, 840, 576, 960, 768, 960, 864, 1152, 864, 1368, 1080, 1344, 1152, 1680, 1152, 1848, 1440, 1728, 1584, 2208, 1536
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OFFSET
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1,2
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COMMENTS
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Number of points in the bicyclic group Z/mZ x Z/mZ whose order is exactly m. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Mar 14 2006
A000056(n)=n*a(n). - Michael Somos Mar 20 2004
Number of irreducible fractions among {(u+v*i)/n:1<=u,v<=n} with i=sqrt(-1), where a fraction (u+v*i)/n is called irreducible iff GCD(u,v,n)=1. - Reinhard Zumkeller, Aug 20 2005
The weight of the n-th polynomial for the analog of cyclotomic polynomials for elliptic divisibility sequences. That is, let weight of b1 = 1, b2 = 3, b3 = 8, b4 = 12 and let e1 = b1, e2 = b2*b1, e3 = b3*b1, e4 = b2*b4*b1, e5 = (b2^4*b4 - b3^3)*b1 = b5*e1 and so on be an elliptic divisibility sequence. Then weight of e2 = 4, e3 = 9, e4 = 16, e5 = 25, where weight of en is n^2 in general, while weight of bn is a(n). - Michael Somos Aug 12 2008
J_2(n) divides J_{2k}(n). J_2(n) counts the number of 2-tuples (x1,x2), such that 1<=x1, x2<=n and GCD(x1,x2,n)=1 [Enrique Pérez Herrero, 5 Mar 2011].
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REFERENCES
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Sukumar Das Adhikari and A. Sankaranarayanan - On an error term related to the Jordan totient function Jk(n) Journal of Number Theory Volume 34, Issue 2, February 1990, Pages 178-188.
Dorin Andrica and Mihai Piticari, On Some Extensions Of Jordan's Arithmetic Functions, Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics - ICTAMI 2003, Alba Iulia.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
L. E. Dickson (1919, repr. 1971). History of the Theory of Numbers I. Chelsea. p. 147.
F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
P. J. McCarthy, Introduction to Arithmetical Functions, Universitext, Springer, New York, NY, USA, 1986.
Nittiya Pabhapote and Vichian Laohakosol, Combinatorial Aspects of the Generalized Euler’s Totient, International Journal of Mathematics and Mathematical Sciences, Volume 2010, Article ID 648165.
G. Polya and G. Szego, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 1, Section 6, Problem 64.
M. Ram Murty (2001). Problems in Analytic Number Theory. Graduate Texts in Mathematics. 206. Springer-Verlag. p. 11.
Wolfgang Schramm, The Fourier transform of functions of the greatest common divisor, Electronic Journal of Combinatorial Number Theory A50 (8(1)), 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Thajoddin, S. Vangipuram - A Note On Jordan's Totient Function, Indian J. Pure Appl. Math, 1988
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
N. J. A. Sloane, Transforms
Wikipedia, Jordan's totient function
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FORMULA
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Moebius transform of squares.
Multiplicative with a(p^e) = p^(2e)-p^(2e-2). - Vladeta Jovovic, Jul 26 2001
a(n) = sum(d|n, d^2*mu(n/d)). - Benoit Cloitre, Apr 05 2002
a(n) = Sum(phi(d)*phi(n/d)*n/d: d divides n); Sum(a(d): d divides n) = n^2. - Reinhard Zumkeller, Aug 20 2005
Dirichlet generating function: zeta(s-2)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = a(n^2)/n^2. [Enrique Pérez Herrero, Sep 14 2010]
a(n) = A000010(n)*A001615(n).
if n>1 then 1>a(n)/n^2>1/zeta(2) [From Enrique Pérez Herrero, Jul 14 2011]
a(n) = sum_{d|n} phi(n^2/d)*mu(d)^2). - Enrique Pérez Herrero, Jul 24 2012
a(n) = Sum_{k=1..n} GCD(k,n)^2 * Cos(2*Pi*k/n). - Enrique Pérez Herrero, Jan 18 2013
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EXAMPLE
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a(4) = 12 because the divisors of 4 being 1, 2, 4, we find that phi(1)phi(4/1)(4/1) = 8, phi(2)phi(4/2)(4/2) = 2, phi(4)phi(4/4)(4/4) = 2 and 8 + 2 + 2 = 12.
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MAPLE
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J := proc(n, k) local i, p, t1, t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end; # (with k = 2)
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MATHEMATICA
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jordanTotient[n_, k_:1] := DivisorSum[n, #^k*MoebiusMu[n/#] &] /; (n > 0) && IntegerQ[n]; Table[jordanTotient[n, 2], {n, 48}] (* Enrique Perez Herrero (psychgeometry (AT)gmail.com), Sep 14 2010 *)
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PROG
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(PARI) a(n)=if(n<1, 0, sumdiv(n, d, d^2*moebius(n/d)))
(Haskell)
a007434 n = sum $ zipWith3 (\x y z -> x * y * z)
tdivs (reverse tdivs) (reverse divs)
where divs = a027750_row n; tdivs = map a000010 divs
-- Reinhard Zumkeller, Nov 24 2012
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CROSSREFS
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Cf. A000290. Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).
A115000(n) = a(n) / 24 unless n<5. - Michael Somos Aug 12 2008
Cf. A027750.
Sequence in context: A169923 A158022 A209934 * A128303 A123906 A065970
Adjacent sequences: A007431 A007432 A007433 * A007435 A007436 A007437
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Thanks to Michael Somos for catching an error in this sequence.
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STATUS
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approved
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