A194533 Jordan function ratio J_8(n)/J_2(n).

1, 85, 820, 5440, 16276, 69700, 120100, 348160, 597780, 1383460, 1786324, 4460800, 4855540, 10208500, 13346320, 22282240, 24221380, 50811300, 47176564, 88541440, 98482000, 151837540, 148316260, 285491200, 254312500, 412720900, 435781620, 653344000, 595531444

Offset: 1

Author

R. J. Mathar, Aug 28 2011

Keywords

• nonn
• mult
• easy

Crossrefs

Cf. A001014, A007434, A065958, A065960, A069093.

Programs

Mathematica

f[p_, e_] := p^(6*(e - 1))*(p^2 + 1)*(p^4 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 01 2022 *)

Formula

a(n) = A069093(n)/A007434(n) = A065960(n) * A065958(n).

Multiplicative with a(p^e) = p^(6*(e-1))*(p^2+1)*(p^4+1), e>0.

Dirichlet g.f.: zeta(s-6)*Product_{primes p} (1+p^(4-s)+p^(2-s)+p^(-s)).

Dirichlet convolution of A001014 with the multiplicative sequence 1, 21, 91, 0, 651, 1911, 2451, 0, 0, 13671, 14763, 0, 28731, 51471...

Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = Product_{primes p} (1 + 1/p^3 + 1/p^5 + 1/p^7) = 1.22847463998021088097249049512949441921891884186337179613337753... - Vaclav Kotesovec, Dec 18 2019