A160897 - OEIS
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, 580865013, 1115624896
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
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
Multiplicative with a(p^e) = p^(6e-6) * (p^7-1) / (p-1).
For squarefree n, a(n) = A000203(n^6). (End)
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)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^7). - Ridouane Oudra, Apr 01 2025
MAPLE
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)) ;
MATHEMATICA
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