A160897 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 8.

1, 127, 1093, 8128, 19531, 138811, 137257, 520192, 796797, 2480437, 1948717, 8883904, 5229043, 17431639, 21347383, 33292288, 25646167, 101193219, 49659541, 158747968, 150021901, 247487059, 154764793, 568569856, 305171875, 664088461

Offset

1,2

Comments

a(n) is the number of lattices L in Z^7 such that the quotient group Z^7 / L is C_n. - Álvar Ibeas, Oct 30 2015

Links

Enrique Pérez Herrero and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..5000 from Enrique Pérez Herrero)

Jin Ho Kwak and Jaeun Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.

Index to Jordan function ratios J_k/J_1.

Formula

a(n) = J_7(n)/J_1(n) = J_7(n)/phi(n) = A069092(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 27 2010

From Álvar Ibeas, Oct 30 2015: (Start)

Multiplicative with a(p^e) = p^(6e-6) * (p^7-1) / (p-1).

For squarefree n, a(n) = A000203(n^6). (End)

From Amiram Eldar, Nov 08 2022: (Start)

Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 + (p^6-1)/((p-1)*p^7)) = 0.2761554804... .

Sum_{k>=1} 1/a(k) = zeta(6)*zeta(7) * Product_{p prime} (1 - 2/p^7 + 1/p^13) = 1.008982290854... . (End)

Maple

A160897 := proc(n)
    add(numtheory[mobius](n/d)*d^7, d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n", n, A160897(n)) ;
end do: # R. J. Mathar, Mar 14 2016

Mathematica

A160897[n_]:=DivisorSum[n, MoebiusMu[n/# ]*#^(8 - 1)/EulerPhi[n] &] (* Enrique Pérez Herrero, Oct 27 2010 *)
f[p_, e_] := p^(6*e - 6) * (p^7-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

Prog

(PARI) vector(30, n, sumdiv(n^6, d, if(ispower(d, 7), moebius(sqrtnint(d, 7))*sigma(n^6/d), 0))) \\ Altug Alkan, Oct 30 2015
(PARI) a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i, 1] = p^(6*f[i, 2]-6)*(1+p+p^2+p^3+p^4+p^5+p^6); f[i, 2] = 1; ); factorback(f); } \\ Michel Marcus, Nov 12 2015

Crossrefs

Column 7 of A263950.

Cf. A000010, A000203, A013664, A013665, A069092.

Sequence in context: A196658 A077361 A225148 * A038994 A068023 A194257

Adjacent sequences: A160894 A160895 A160896 * A160898 A160899 A160900

Keyword

nonn,mult

Author

N. J. A. Sloane, Nov 19 2009

Extensions

Definition corrected by Enrique Pérez Herrero, Oct 27 2010

Status

approved