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A069091 Jordan function J_6(n). 9
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

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

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 *)

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]

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

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Last modified December 8 02:07 EST 2019. Contains 329850 sequences. (Running on oeis4.)