cp's OEIS Frontend

A161167 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 17.

%I A161167 #41 Apr 02 2025 17:27:06
%S A161167 1,65535,21523360,2147450880,38146972656,1410533397600,5538821761600,
%T A161167 70367670435840,308836690967520,2499961853010960,4594972986357216,
%U A161167 46220358372556800,55451384098598320,362986684146456000,821051025385244160,2305807824841605120,3041324492229179280
%N A161167 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 17.
%C A161167 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 A161167 Enrique Pérez Herrero, <a href="/A161167/b161167.txt">Table of n, a(n) for n = 1..5000</a>
%H A161167 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 A161167 <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.
%F A161167 a(n) = J_16(n)/J_1(n) = J_16(n)/A000010(n), where J_k is the k-th Jordan totient function.
%F A161167 From _Álvar Ibeas_, Nov 26 2015: (Start)
%F A161167 Multiplicative with a(p^e) = p^(15e-15) * (p^16-1) / (p-1).
%F A161167 For squarefree n, a(n) = A000203(n^15). (End)
%F A161167 From _Amiram Eldar_, Nov 08 2022: (Start)
%F A161167 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 A161167 Sum_{k>=1} 1/a(k) = zeta(15)*zeta(16) * Product_{p prime} (1 - 2/p^16 + 1/p^31) = 1.00001530597583... . (End)
%F A161167 a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^16). - _Ridouane Oudra_, Apr 02 2025
%p A161167 A161167 := proc(n)
%p A161167     add(numtheory[mobius](n/d)*d^16,d=numtheory[divisors](n)) ;
%p A161167     %/numtheory[phi](n) ;
%p A161167 end proc:
%p A161167 for n from 1 to 5000 do
%p A161167     printf("%d %d\n",n,A161167(n)) ;
%p A161167 end do: # _R. J. Mathar_, Mar 15 2016
%t A161167 A161167[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(17-1)/EulerPhi[n]&]; Array[A161167,20]
%t A161167 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 A161167 (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 A161167 (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 A161167 Column 16 of A263950.
%Y A161167 Cf. A000010, A000203, A008683, A013673, A013674.
%K A161167 nonn,mult
%O A161167 1,2
%A A161167 _N. J. A. Sloane_, Nov 19 2009
%E A161167 Definition corrected by _Enrique Pérez Herrero_, Oct 30 2010