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a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 17.
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%I #36 Nov 08 2022 08:08:07

%S 1,65535,21523360,2147450880,38146972656,1410533397600,5538821761600,

%T 70367670435840,308836690967520,2499961853010960,4594972986357216,

%U 46220358372556800,55451384098598320,362986684146456000

%N a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 17.

%C a(n) is the number of lattices L in Z^16 such that the quotient group Z^16 / L is C_n. - _Álvar Ibeas_, Nov 26 2015

%H Enrique Pérez Herrero, <a href="/A161167/b161167.txt">Table of n, a(n) for n = 1..5000</a>

%H Jin Ho Kwak and Jaeun Lee, <a href="https://doi.org/10.1142/9789812799890_0005">Enumeration of graph coverings, surface branched coverings and related group theory</a>, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.

%H <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.

%F a(n) = J_16(n)/J_1(n) = J_16(n)/A000010(n), where J_k is the k-th Jordan totient function.

%F From _Álvar Ibeas_, Nov 26 2015: (Start)

%F Multiplicative with a(p^e) = p^(15e-15) * (p^16-1) / (p-1).

%F For squarefree n, a(n) = A000203(n^15). (End)

%F From _Amiram Eldar_, Nov 08 2022: (Start)

%F Sum_{k=1..n} a(k) ~ c * n^16, where c = (1/16) * Product_{p prime} (1 + (p^15-1)/((p-1)*p^16)) = 0.1214735403... .

%F Sum_{k>=1} 1/a(k) = zeta(15)*zeta(16) * Product_{p prime} (1 - 2/p^16 + 1/p^31) = 1.00001530597583... . (End)

%p A161167 := proc(n)

%p add(numtheory[mobius](n/d)*d^16,d=numtheory[divisors](n)) ;

%p %/numtheory[phi](n) ;

%p end proc:

%p for n from 1 to 5000 do

%p printf("%d %d\n",n,A161167(n)) ;

%p end do: # _R. J. Mathar_, Mar 15 2016

%t A161167[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(17-1)/EulerPhi[n]&]; Array[A161167,20]

%t f[p_, e_] := p^(15*e - 15) * (p^16-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, Nov 08 2022 *)

%o (PARI) vector(100, n, sumdiv(n^15, d, if(ispower(d, 16), moebius(sqrtnint(d, 16))*sigma(n^15/d), 0))) \\ _Altug Alkan_, Nov 26 2015

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^16 - 1)*f[i,1]^(15*f[i,2] - 15)/(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 08 2022

%Y Column 16 of A263950.

%Y Cf. A000010, A000203, A013673, A013674.

%K nonn,mult

%O 1,2

%A _N. J. A. Sloane_, Nov 19 2009

%E Definition corrected by _Enrique Pérez Herrero_, Oct 30 2010