User:Enrique Pérez Herrero/Jordan

Jordan's Totient Function

Definition-1

• $J_{k}(n)$ counts the number of k-tuples $(x_{1},x_{2},\dots ,x_{k})$ such that $1\leq x_{1},x_{2},\dots ,x_{k}\leq n$ and $GCD(x_{1},x_{2},\dots ,x_{k},n)=1$

Definition-2

• $\displaystyle J_{k}(n)=\sum _{d|n}{\mu (n/d)\cdot d^{k}}$

Jordan Totient Functions in OEIS

Jordan's Totient Function

Function	Sequence Id
$\displaystyle J_{1}(n)=\phi (n)$	A000010
$\displaystyle J_{2}(n)$	A007434
$\displaystyle J_{3}(n)$	A059376
$\displaystyle J_{4}(n)$	A059377
$\displaystyle J_{5}(n)$	A059378
$\displaystyle J_{6}(n)$	A069091
$\displaystyle J_{7}(n)$	A069092
$\displaystyle J_{8}(n)$	A069093
$\displaystyle J_{9}(n)$	A069094
$\displaystyle J_{10}(n)$	A069095

Formulas:

• $\displaystyle J_{k}(n)=n^{k}\prod _{p|n}(1-p^{-k})$

• $\displaystyle \sum _{d|n}{J_{k}(d)}=n^{k}$

• $\displaystyle J_{k}(n)=\sum _{d|n}{\mu (n/d)\cdot d^{k}}$

• $\displaystyle \sum _{i=1}^{n}{J_{k}(i)\cdot {\bigg \lfloor }{\frac {n}{i}}{\bigg \rfloor }}=\sum _{i=1}^{n}{i^{k}}$

• $\displaystyle J_{k}(n)=\sum _{j=1}^{n}{{GCD(n,j)}^{k}\cdot \cos {2\pi {\frac {j}{n}}}}$

• $\displaystyle J_{k}(p^{\alpha })=p^{k\cdot \alpha }-p^{k\cdot (\alpha -1)}$

• $\displaystyle J_{k}(2)=2^{k}-1=M_{k}$ , Sequence A000225

• $\displaystyle 1>{\frac {J_{k}(n)}{n^{k}}}>{\frac {1}{\zeta (k)}}$ $,(n>1)$

• $\displaystyle J_{n}(n)$, Sequence A067858

• $\displaystyle {\frac {J_{4}(n)}{\psi (n)\cdot \phi (n)}}={\frac {J_{4}(n)}{J_{2}(n)}}$, Sequence A065958

• $\displaystyle {\frac {J_{6}(n)}{J_{3}(n)}}$, Sequence A065959

• $\displaystyle {\frac {J_{8}(n)}{J_{4}(n)}}$, Sequence A065960

• $\displaystyle {\frac {J_{10}(n)}{J_{5}(n)}}$, Sequence A351300

• $\displaystyle {\frac {J_{12}(n)}{J_{6}(n)}}$, Sequence A351301

• $\displaystyle {\frac {J_{14}(n)}{J_{7}(n)}}$, Sequence A351302

• $\displaystyle {\frac {J_{16}(n)}{J_{8}(n)}}$, Sequence A351303

Eulerphi divides Jordan´s Totient Function

Function	Sequence Id
$\displaystyle {\frac {J_{2}(n)}{\phi (n)}}=\psi (n)$ [1]	A001615
$\displaystyle {\frac {J_{3}(n)}{\phi (n)}}$	A160889
$\displaystyle {\frac {J_{4}(n)}{\phi (n)}}$	A160891
$\displaystyle {\frac {J_{5}(n)}{\phi (n)}}$	A160893
$\displaystyle {\frac {J_{6}(n)}{\phi (n)}}$	A160895
$\displaystyle {\frac {J_{7}(n)}{\phi (n)}}$	A160897
$\displaystyle {\frac {J_{8}(n)}{\phi (n)}}$	A160908
$\displaystyle {\frac {J_{9}(n)}{\phi (n)}}$	A160953
$\displaystyle {\frac {J_{10}(n)}{\phi (n)}}$	A160957
$\displaystyle {\frac {J_{11}(n)}{\phi (n)}}$	A160960
$\displaystyle {\frac {J_{12}(n)}{\phi (n)}}$	A160972
$\displaystyle {\frac {J_{13}(n)}{\phi (n)}}$	A161010
$\displaystyle {\frac {J_{14}(n)}{\phi (n)}}$	A161025
$\displaystyle {\frac {J_{15}(n)}{\phi (n)}}$	A161139
$\displaystyle {\frac {J_{16}(n)}{\phi (n)}}$	A161167
$\displaystyle {\frac {J_{17}(n)}{\phi (n)}}$	A161213

• Definitions Corrected: A160891,A160893,A160895,A160897,A160908,A160953,A160957,A160960,A160972,A161010,A161025,A161139,A161167,A161213.

• If $\displaystyle d|k$ then $\displaystyle J_{d}(n)|J_{k}(n)$

• $\displaystyle \psi (n)|J_{k}(n)$ iff $k$ is even.

• $\displaystyle J_{k}(n)\cdot {rad(n)}^{k}=n^{k}\cdot J_{k}(rad(n))$

• $\displaystyle {\sqrt[{k}]{\frac {J_{k}(n)}{J_{k}(rad(n))}}}={\frac {n}{rad(n)}}$, Sequence A003557

• ${\frac {J_{2}(n^{2})}{n}}={\frac {J_{2}(n^{3})}{n^{3}}}=n\cdot J_{2}(n)$, Sequence A000056

• $J_{2}(n)={\frac {J_{2}(n^{2})}{n^{2}}}$

• ${\frac {J_{3}(n^{2})}{n}}$, Sequence: A115224

Arithmetical Matrix Determinants related to Jordan's Functions:

Generalization of Le Paige´s Theorem:

• $det{{\bigg [}{GCD(i,j)}^{k}{\bigg ]}}_{1\leq i,j\leq n}=\prod _{i=1}^{n}{J_{k}(i)}$

Jordan's Function Determinants

Function	Sequence Id
$\prod _{i=1}^{n}{\psi _{1}(i)}$	A175836
$\prod _{i=1}^{n}{J_{1}(i)}$	A001088
$\prod _{i=1}^{n}{J_{2}(i)}$	A059381
$\prod _{i=1}^{n}{J_{3}(i)}$	A059382
$\prod _{i=1}^{n}{J_{4}(i)}$	A059383
$\prod _{i=1}^{n}{J_{5}(i)}$	A059384
$\prod _{i=1}^{n}{J_{6}(i)}$	A239672

Array of values of Jordan function $J_{k}(n)$ read by antidiagonals

• Sequences: A059379 and A059380

Mathematica Code:

(* Function Definitions *)
JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n]

A000010[n_IntegerQ]:=EulerPhi[n];
A007434[n_IntegerQ]:=JordanTotient[n,2];
A059376[n_IntegerQ]:=JordanTotient[n,3];
A059377[n_IntegerQ]:=JordanTotient[n,4];
A059378[n_IntegerQ]:=JordanTotient[n,5];
A069091[n_IntegerQ]:=JordanTotient[n,6];
A069092[n_IntegerQ]:=JordanTotient[n,7];
A069093[n_IntegerQ]:=JordanTotient[n,8];
A069094[n_IntegerQ]:=JordanTotient[n,9];
A069095[n_IntegerQ]:=JordanTotient[n,10];

(* Related Sequences *)
A067858[n_]:=JordanTotient[n,n];
A160889[n_]:=JordanTotient[n,3]/EulerPhi[n];
 
(* Adding Help *)
JordanTotient::usage="JordanTotient[n,k] is a generalization of EulerPhi[n].";

A000010::usage="A000010[n]: Euler totient function phi(n): count numbers <= n and prime to n.";
A007434::usage="A007434[n]: Jordan function J_2(n) (a generalization of phi(n)).";
A059376::usage="A059376[n]: Jordan function J_3(n).";
A059377::usage="A059377[n]: Jordan function J_4(n).";
A059378::usage="A059378[n]: Jordan function J_5(n).";
A069091::usage="A069091[n]: Jordan function J_6(n).";
A069092::usage="A069092[n]: Jordan function J_7(n).";
A069093::usage="A069093[n]: Jordan function J_8(n).";
A069094::usage="A069094[n]: Jordan function J_9(n).";
A069095::usage="A069095[n]: Jordan function J_10(n)."; 

A067858::usage="A067858[n]: J_n(n), where J is the Jordan function, J_n(n) = n^n product{p|n}(1 - 1/p^n), the product
is over the distinct primes, p, dividing n.";

A160889::usage="A160889: Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 4."; 

(* Attributes *)
SetAttributes[{JordanTotient,A000010,A007434,A059376,A059377,A059378,A069091,A069092,A069093,A069094,A069095},Listable];

• Download Full Mathematica Package
• 09/19/10 More functions added

PARI/GP Code:

\\ Function definitions: 

jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
dedekindpsi(n)=jordantot(n,2)/eulerphi(n);

A000010(n)=eulerphi(n);
A007434(n)=jordantot(n,2);
A059376(n)=jordantot(n,3);
A059377(n)=jordantot(n,4);
A059378(n)=jordantot(n,5);
A069091(n)=jordantot(n,6);
A069092(n)=jordantot(n,7);
A069093(n)=jordantot(n,8);
A069094(n)=jordantot(n,9);
A069095(n)=jordantot(n,10);
A067858(n)=jordantot(n,n); 

A067858(n)=jordantot(n,n);
A000225(n)=jordantot(2,n);
A001615(n)=dedekindpsi(n);
A160889(n)=jordantot(n,3)/jordantot(n);
A160891(n)=jordantot(n,4)/jordantot(n);
A160893(n)=jordantot(n,5)/jordantot(n);
A160895(n)=jordantot(n,6)/jordantot(n);
A160897(n)=jordantot(n,7)/jordantot(n);
A160908(n)=jordantot(n,8)/jordantot(n);
A160960(n)=jordantot(n,9)/jordantot(n);
A160957(n)=jordantot(n,10)/jordantot(n);
A160960(n)=jordantot(n,11)/jordantot(n);
A160972(n)=jordantot(n,12)/jordantot(n);
A161010(n)=jordantot(n,13)/jordantot(n);
A161025(n)=jordantot(n,14)/jordantot(n);
A161139(n)=jordantot(n,15)/jordantot(n);
A161167(n)=jordantot(n,16)/jordantot(n);
A161213(n)=jordantot(n,17)/jordantot(n); 

A065958(n)=jordantot(n,4)/jordantot(n,2);
A065959(n)=jordantot(n,6)/jordantot(n,3);
A065960(n)=jordantot(n,8)/jordantot(n,4);

• Download complete PARI/GP code.

Links:

• Peter Luschny, OEIS Wiki, Euler Totient
• Daniel Forgues OEIS Wiki, Dedekind psi function
• Jordan's Totient function-Wikipedia
• Dedekind Psi function-Wikipedia
• PlanetMath: Jordan's totient function
• L. E. Dickson (1919, repr.1971). History of the Theory of Numbers I. Chelsea. p. 147. ISBN 0-8284-0086-5.
• M. Ram Murty (2001). Problems in Analytic Number Theory. Graduate Texts in Mathematics. 206. Springer-Verlag. p. 11. ISBN 0387951431.
• S Thajoddin, S Vangipuram - A NOTE ON JORDAN'S TOTIENT FUNCTION -Indian J. Pure Appl. Math, 1988
• 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
• Matthew Holden, Michael Orrison, Michael Vrable, Yet Another Generalization of Euler's Totient Function
• 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
• 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, 15 pages, doi:10.1155/2010/648165
• P. J. McCarthy, Introduction to Arithmetical Functions, Universitext, Springer, New York, NY, USA, 1986.