A177821 a(n) gives the number of nonisomorphic connected compact Lie groups of dimension n which are simple products.
1, 1, 1, 3, 3, 3, 6, 6, 8, 12, 14, 18, 23, 27, 34, 43, 52, 62, 79, 93, 109, 138, 159, 187, 236, 270, 316, 385, 442, 517, 619, 716, 833, 980, 1132, 1308, 1533, 1758, 2027, 2370, 2703, 3095, 3594, 4081, 4668, 5397, 6125, 6970, 8007, 9065, 10281, 11753, 13289, 15036, 17120, 19305, 21788, 24690, 27768, 31294, 35381, 39690, 44591, 50261, 56267, 63047
Offset: 0
Keywords
Examples
For n=0, the trivial group is the only such group. For n=8, the 8 Lie groups are U(1)^8, U(1)^5 x SU(2), U(1)^5 x SO(3), U(1)^2 x SU(2)^2, U(1)^2 x SU(2) x SO(3), U(1)^2 x SO(3)^2, SU(3) and SU(3)/3.
Crossrefs
See also A178176 for enumeration of the simple factors giving these counts.
Formula
G.f.: 1/((1-x)*(1-x^3)^2*(1-x^8)^2*(1-x^10)^2*(1-x^14)*...) = (1/(1-x)) * Product_{k>=0} (1-x^k)^A178176(k) with (1-x^k)^0 taken to be 1.
Extensions
a(28) and following corrected by Andrea Aveni, Mar 22 2025
Comments