A194532 Jordan function ratio J_6(n)/J_2(n).

1, 21, 91, 336, 651, 1911, 2451, 5376, 7371, 13671, 14763, 30576, 28731, 51471, 59241, 86016, 83811, 154791, 130683, 218736, 223041, 310023, 280371, 489216, 406875, 603351, 597051, 823536, 708123, 1244061, 924483, 1376256, 1343433, 1760031, 1595601, 2476656, 1875531, 2744343

Offset: 1

Author

R. J. Mathar, Aug 28 2011

Keywords

• nonn
• mult
• easy

Comments

Dirichlet convolution of A000583 with the multiplicative function which starts 1, 5, 10, 0, 26, 50, 50, 0, 0, 130, 122, 0, 170, 250, 260, 0, 290,..

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007434, A065959, A069091.

Programs

Maple

f:= proc(n) local t;
     mul(t[1]^(4*(t[2]-1))*((t[1]^2+1)^2-t[1]^2),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jun 14 2016

Mathematica

JordanTotient[n_, k_: 1] := DivisorSum[n, #^k MoebiusMu[n/#] &] /; (n > 0) && IntegerQ@ n; Table[JordanTotient[n, 6]/JordanTotient[n, 2], {n, 12}] (* Michael De Vlieger, Jun 14 2016, after Enrique Pérez Herrero at A065959 *)
f[p_, e_] := p^(4*(e-1))*(p^2+p+1)*(p^2-p+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(4*(f[i,2]-1))*(f[i,1]^2+f[i,1]+1)*(f[i,1]^2-f[i,1]+1));} \\ Amiram Eldar, Nov 05 2022

Formula

a(n) = A069091(n)/A007434(n).

Multiplicative with a(p^e) = p^(4*(e-1))*(p^2+p+1)*(p^2-p+1), e>0.

Dirichlet g.f.: zeta(s-4)*product_{primes p} (1+p^(2-s)+p^(-s)).

Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = Product_{primes p} (1 + 1/p^3 + 1/p^5) = 1.2196771388395597011492820972459808778277319864216893177353903924... - Vaclav Kotesovec, Dec 18 2019

Sum_{n>=1} 1/a(n) = (Pi^8/14175) * Product_{p prime} (1 + 1/p^2 + 1/p^4 - 1/p^6 - 1/p^8) = 1.06469274411... . - Amiram Eldar, Nov 05 2022