login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A069091 Jordan function J_6(n). 10
1, 63, 728, 4032, 15624, 45864, 117648, 258048, 530712, 984312, 1771560, 2935296, 4826808, 7411824, 11374272, 16515072, 24137568, 33434856, 47045880, 62995968, 85647744, 111608280, 148035888, 187858944, 244125000, 304088904, 386889048 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Enrique Pérez Herrero, Sep 14 2010: (Start)

a(n) is the Moebius transform of n^6.

Note that J_2(n), J_3(n), eulerphi(n) and psi(n) divides a(n), this sequences

are: A007434(n), A059376(n), A000010(n) and A001615(n) respectively. (End)

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

LINKS

Enrique Pérez Herrero, Table of n, a(n) for n=1..2000

Wikipedia, Jordan's totient function.

FORMULA

a(n) = Sum_{d|n} d^6*mu(n/d).

Multiplicative with a(p^e) = p^(6e)-p^(6(e-1)).

Dirichlet generating function: zeta(s-6)/zeta(s). - Ralf Stephan, Jul 04 2013

a(n) = n^6*Product_{distinct primes p dividing n} (1-1/p^6). - Tom Edgar, Jan 09 2015

Sum_{k=1..n} a(k) ~ n^7 / (7*zeta(7)). - Vaclav Kotesovec, Feb 07 2019

From Amiram Eldar, Oct 12 2020: (Start)

lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^6 = 1/zeta(7).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^6/(p^6-1)^2) = 1.0175973008... (End)

MATHEMATICA

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

A069091[n_IntegerQ]:=JordanTotient[n, 6]; (* Enrique Pérez Herrero, Sep 14 2010 *)

f[p_, e_] := p^(6*e) - p^(6*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

PROG

(PARI) for(n=1, 100, print1(sumdiv(n, d, d^6*moebius(n/d)), ", "))

CROSSREFS

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).

Cf. A065959. [Enrique Pérez Herrero, Sep 14 2010]

Cf. A013665.

Sequence in context: A221968 A115152 A284953 * A123866 A024004 A284927

Adjacent sequences:  A069088 A069089 A069090 * A069092 A069093 A069094

KEYWORD

easy,nonn,mult

AUTHOR

Benoit Cloitre, Apr 05 2002

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 30 20:11 EDT 2020. Contains 338090 sequences. (Running on oeis4.)