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A178176 a(n) is the number of central quotients of simple compact Lie groups of dimension n.

%I A178176 #45 Apr 05 2025 09:02:37
%S A178176 0,0,2,0,0,0,0,2,0,2,0,0,0,1,3,0,0,0,0,0,4,0,0,2,0,0,0,3,0,0,0,0,0,0,
%T A178176 4,4,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,1,0,0,4,0,0,0,0,0,0,0,4,0,0,4,0,0,
%U A178176 0,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4
%N A178176 a(n) is the number of central quotients of simple compact Lie groups of dimension n.
%H A178176 Andrea Aveni, <a href="/A178176/b178176.txt">Table of n, a(n) for n = 1..1000</a>
%H A178176 Math Overflow, <a href="https://mathoverflow.net/questions/489474/number-of-compact-connected-lie-groups-of-given-dimension">Number of compact connected Lie groups of given dimension</a>
%H A178176 nLab, <a href="https://ncatlab.org/nlab/show/semi-spin+group">semi-spin group</a>
%e A178176 a(3) = 2 since the 3-dimensional SU(2) has two central quotients: SU(2) and SU(2)/2 = SO(3).
%e A178176 a(28) = 3 and not 4 since, because of triality for Spin(8), the semi-spin group HSpin(8) is isomorphic to SO(8). Thus, the only groups are Spin(8), SO(8), PSO(8). See the nLab link.
%e A178176 The unusually large value a(78) = 6 is due to Spin(13), SO(13), Sp(6), PSp(6), E_6, E_6/Z3 all of dimension 78.
%o A178176 (R)
%o A178176 Number.Divisors=function(n){
%o A178176   out=c()
%o A178176   for(j in 1:n){if(n%%j==0){out=c(out,j)}}
%o A178176   return(length(out))
%o A178176 }
%o A178176 a178176=function(n){
%o A178176   kSU=sqrt(n+1)
%o A178176   kSO=(sqrt(8*n+1)+1)/2
%o A178176   kSp=(sqrt(8*n+1)-1)/4
%o A178176   a=0
%o A178176   if(n %in% c(14,52,248)){a=a+1} # G2, F4, E8 with center Z1
%o A178176   if(n %in% c(78,133)){a=a+2} # E7 with center Z2, E6 with center Z3
%o A178176   if(kSp%%1==0 & kSp>=2){a=a+2} # Sp(k), PSp(k)
%o A178176   if(kSU%%1==0 & kSU>=2){a=a+Number.Divisors(kSU)} # SU(n)/Zd
%o A178176   if(kSO%%1==0 & kSO>=7 & kSO!=8){
%o A178176     if(kSO%%2!=0){a=a+2} # Spin(k), SO(k)
%o A178176     if(kSO%%2==0 & kSO%%4!=0){a=a+3} # Spin(k), SO(k), PSO(k)
%o A178176     if(kSO%%4==0){a=a+4} # Spin(k), SO(k), HSpin(k), PSO(k)
%o A178176   }
%o A178176   if(n==28){a=3} # Because of Triality: Spin(8), HSpin(8)=SO(8), PSO(8)
%o A178176   return(a)
%o A178176 } # _Andrea Aveni_, Mar 23 2025
%K A178176 nonn
%O A178176 1,3
%A A178176 _Andrew Rupinski_, Dec 18 2010
%E A178176 a(28) corrected by _Andrea Aveni_, Mar 23 2025