A070943 Commuting elements: number of ordered pairs g, h in the group GL(2,Z_n) such that gh = hg.

1, 18, 384, 1344, 11520, 6912, 96768, 92160, 303264, 207360, 1584000, 516096, 4402944, 1741824, 4423680, 6094848, 22560768, 5458752, 44323200, 15482880, 37158912, 28512000, 141064704, 35389440, 186000000, 79252992, 226748160, 130056192, 572947200, 79626240

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 12 2003

Keywords

• mult
• nonn

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000010, A000203, A000252, A062354.

Programs

Mathematica

a[n_] := n^4*DivisorSigma[1, n]*EulerPhi[n]*Product[(1-1/p^2)*(1-1/p), {p, FactorInteger[n][[All, 1]]}]; a[1]=1; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, May 02 2013, after Eric M. Schmidt *)

Sage

def A070943(n) : return Integer(n^4 * sigma(n) * euler_phi(n) * prod((1-1/p^2)*(1-1/p) for (p,m) in factor(n))) # Eric M. Schmidt, May 02 2013

Formula

a(n) = A000252(n) * A062354(n).

a(n) = n^4 * Product_{p prime, p|n} (1-1/p^2)*(1-1/p) * sigma(n)*phi(n).

From Amiram Eldar, Oct 30 2022: (Start)

Multiplicative with a(p^e) = (p^(e+1)-1) * (p-1)^2 * (p+1) * p^(5*e-4).

Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 - 1/p^2 - 2/p^3 + 3/p^4 - 1/p^5) = 0.07103214283... . (End)

Extensions

More terms from Benoit Cloitre, Sep 13 2003

More terms from Eric M. Schmidt, May 02 2013