login
This site is supported by donations to The OEIS Foundation.

 

Logo

Many excellent designs for a new banner were submitted. We will use the best of them in rotation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007434 Jordan function J_2(n) (a generalization of phi(n)).
(Formerly M2717)
36
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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 if and only if 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, Mar 05 2011

REFERENCES

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

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

MathOverFlow, Averages of euler-phi function and similar

N. J. A. Sloane, Transforms

Wikipedia, Jordan's totient function

FORMULA

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

Dirichlet inverse of A046970. - Michael Somos, Jan 11 2014

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). - 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

a(1) + a(2) + ... + a(n) ~ 1/(3*zeta(3))*n^3 + O(n^2). Lambert series sum {n >= 1} a(n)*x^n/(1 - x^n) = x*(1 + x)/(1 - x)^3. - Peter Bala, Dec 23 2013

n * a(n) = A000056(n). - Michael Somos, Mar 20 2004

a(n) = 24 * A115000(n) unless n<5. - Michael Somos, Aug 12 2008

EXAMPLE

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.

G.f. = x + 3*x^2 + 8*x^3 + 12*x^4 + 24*x^5 + 24*x^6 + 48*x^7 + 48*x^8 + 72*x^9 + ...

MAPLE

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)

MATHEMATICA

jordanTotient[n_, k_:1] := DivisorSum[n, #^k*MoebiusMu[n/#] &] /; (n > 0) && IntegerQ[n]; Table[jordanTotient[n, 2], {n, 48}] (* Enrique Pérez Herrero, Sep 14 2010 *)

a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ n/d], {d, Divisors @ n}]] (* Michael Somos, Jan 11 2014 *)

a[ n_] := If[ n < 2, Boole[ n == 1], n^2 (Times @@ ((1 - 1/#[[1]]^2) & /@ FactorInteger @ n))] (* Michael Somos, Jan 11 2014 *)

PROG

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d^2 * moebius(n / d)))} /* Michael Somos, Mar 20 2004 */

(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

(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, (1 - X) / (1 - X*p^2))[n])} /* Michael Somos, Jan 11 2014 */

CROSSREFS

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).

Cf. A000056, A027750, A046970, A115000.

Sequence in context: A169923 A158022 A209934 * A128303 A123906 A065970

Adjacent sequences:  A007431 A007432 A007433 * A007435 A007436 A007437

KEYWORD

nonn,easy,nice,mult,look

AUTHOR

N. J. A. Sloane

EXTENSIONS

Thanks to Michael Somos for catching an error in this sequence.

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified April 21 01:47 EDT 2014. Contains 240824 sequences.