A161213 - OEIS

%I #37 Nov 08 2022 08:08:16

%S 1,131071,64570081,8589869056,190734863281,8463265086751,

%T 38771752331201,562945658454016,2779530261754401,24999809265103951,

%U 50544702849929377,554648540725313536,720867993281778161,5081852349802846271

%N a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 18.

%C a(n) is the number of lattices L in Z^17 such that the quotient group Z^17 / L is C_n. - _Álvar Ibeas_, Nov 26 2015

%H Enrique Pérez Herrero, <a href="/A161213/b161213.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_17(n)/A000010(n), where J_17 is the 17th Jordan totient function.

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

%F Multiplicative with a(p^e) = p^(16e-16) * (p^17-1) / (p-1).

%F For squarefree n, a(n) = A000203(n^16). (End)

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

%F Sum_{k=1..n} a(k) ~ c * n^17, where c = (1/17) * Product_{p prime} (1 + (p^16-1)/((p-1)*p^17)) = 0.1143286202... .

%F Sum_{k>=1} 1/a(k) = zeta(16)*zeta(17) * Product_{p prime} (1 - 2/p^17 + 1/p^33) = 1.000007645061593... . (End)

%p A161213 := proc(n)

%p     add(numtheory[mobius](n/d)*d^17,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,A161213(n)) ;

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

%t A161213[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(18-1)/EulerPhi[n]&]; Array[A161213,20]

%t f[p_, e_] := p^(16*e - 16) * (p^17-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, Nov 08 2022 *)

%o (PARI) A161213(n)=sumdiv(n,d,moebius(n/d)*d^17)/eulerphi(n);

%o (PARI) vector(100, n, sumdiv(n^16, d, if(ispower(d, 17), moebius(sqrtnint(d, 16))*sigma(n^16/d), 0))) \\ _Altug Alkan_, Nov 26 2015

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^17 - 1)*f[i,1]^(16*f[i,2] - 16)/(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 08 2022

%Y Column 17 of A263950.

%Y Cf. A000010, A000203, A013674, A013675.

%K nonn,mult

%O 1,2

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

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