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

1,2

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

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

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 *)

Prog

(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

Crossrefs

Cf. A007434, A065959, A069091.

Sequence in context: A259758 A353056 A203173 * A065827 A318036 A326164

Adjacent sequences: A194529 A194530 A194531 * A194533 A194534 A194535

Keyword

nonn,mult,easy

Author

R. J. Mathar, Aug 28 2011

Status

approved