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.
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
KEYWORD
nonn,mult
AUTHOR
N. J. A. Sloane, Nov 19 2009
EXTENSIONS
Definition corrected by Enrique Pérez Herrero, Oct 27 2010
STATUS
approved