A069092 Jordan function J_7(n).

1, 127, 2186, 16256, 78124, 277622, 823542, 2080768, 4780782, 9921748, 19487170, 35535616, 62748516, 104589834, 170779064, 266338304, 410338672, 607159314, 893871738, 1269983744, 1800262812, 2474870590, 3404825446, 4548558848

Offset

1,2

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^7*mu(n/d).

Multiplicative with a(p^e) = p^(7e)-p^(7(e-1)).

Dirichlet generating function: zeta(s-7)/zeta(s). - Ralf Stephan, Jul 04 2013

a(n) = n^7*Product_{distinct primes p dividing n} (1-1/p^7). - Tom Edgar, Jan 09 2015

Sum_{k=1..n} a(k) ~ 4725*n^8 / (4*Pi^8). - Vaclav Kotesovec, Feb 07 2019

From Amiram Eldar, Oct 12 2020: (Start)

lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^7 = 1/zeta(8).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^7/(p^7-1)^2) = 1.0084115178... (End)

O.g.f.: Sum_{n >= 1} mu(n)*A_7(x^n)/(1 - x^n)^8 = x + 127*x^2 + 2186*x^3 + 16256*x^4 + 78124*x^5 + ..., where A_7(x) = x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7 is the 7th Eulerian polynomial. See A008292. - Peter Bala, Jan 31 2022

Mathematica

JordanTotient[n_, k_: 1] := DivisorSum[n, (#^k)*MoebiusMu[n/# ] &] /; (n > 0) && IntegerQ[n]
A069092[n_] := JordanTotient[n, 7]; (* Enrique Pérez Herrero, Nov 02 2010 *)
f[p_, e_] := p^(7*e) - p^(7*(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^7*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. A069091 (J_6), A069092 (J_7), A069093 (J_8), A069094 (J_9), A069095 (J_10). [Enrique Pérez Herrero, Nov 02 2010]

Cf. A013666.

Sequence in context: A022523 A090029 A152726 * A024005 A258808 A321552

Adjacent sequences: A069089 A069090 A069091 * A069093 A069094 A069095

Keyword

easy,nonn,mult

Author

Benoit Cloitre, Apr 05 2002

Status

approved