%I A128644
%S A128644 1,1,1,2,1,2,1,5,2,2,1,5,1,2,1,1,5,1,5,2,2,1,2,2,5,4,1,4,1,1,2,1,1,2,2,
%T A128644 1,6,1,4,2,2,1,2,5,1,5,1,2,2,2,1,1,2,4,1,4,1,5,1,4,1,1,2,3,4,1,6,1,2,1,
%U A128644 1,2,1,1,1,4,2,2,1,1,5,2,1,4,1,2,2,1,1,6,2,1,6,1,5,4,2,1,2,2,1,4,5,1,2
%N A128644 Number of groups of order A037144(n).
%C A128644 Number of groups whose order has at most 3 prime factors.
%C A128644 The groups of these orders (up to A037144(473273456) = 1073741821 in 
               version V2.13-4) form a class contained in the Small Groups Library 
               of MAGMA.
%H A128644 Klaus Brockhaus, <a href="b128644.txt">Table of n, a(n) for n=1..10000</
               a>
%H A128644 MAGMA Documentation, <a href="http://magma.maths.usyd.edu.au/magma/htmlhelp/
               text404.htm">Database of Small Groups</a>
%F A128644 a(n) = A000001(A037144(n)).
%e A128644 A037144(17) = 18 and there are 5 groups of order 18 (A000001(18) = 5), 
               hence a(17) = 5.
%o A128644 (MAGMA) D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [ 
               h: h in [1..130] | h eq 1 or &+[ t[2]: t in Factorization(h) ] le 
               3 ] ];
%o A128644 (PARI) /* based on the formulae from Mitch Harris in A000001 */ {ngoam3pf(n) 
               = local(f, g, nf, ng, p, q, r, qmp, rmp, rmq); f=factor(n); nf=matsize(f)[1]; 
               g=sum(i=1, nf, f[i, 2]); if(g<1, ng=1, if(g>3, ng=-1, if(nf==1, if(f[1, 
               2]==1, ng=1, if(f[1, 2]==2, ng=2, if(f[1, 2]==3, ng=5, ng=-1))), 
               if(nf==2, if(f[1, 2]*f[2, 2]==1, if(gcd(f[1, 1], f[2, 1]-1)==1, ng=1, 
               ng=2), if(f[1, 2]==1, p=f[1, 1]; q=f[2, 1], q=f[1, 1]; p=f[2, 1]); 
               if(p==2&&q%2>0, ng=5, if(q%p==1&&p%2>0, ng=(p+9)/2, if(p==3&&q==2, 
               ng=5, if(p%2>0&&q%2>0&&q%p==p-1, ng=3, if(p>3&&p%q==1&&p%q^2!=1, 
               ng=4, if(p%q^2==1, ng=5, if(q%p!=1&&q%p!=(p-1)&&p%q!=1, ng=2)))))))), 
               p=f[1, 1]; q=f[2, 1]; r=f[3, 1]; qmp=q%p==1; rmp=r%p==1; rmq=r%q==1; 
               if(qmp, if(rmp, if(rmq, ng=p+4, ng=p+2), if(rmq, ng=3, ng=2)), if(rmp, 
               if(rmq, ng=4, ng=2), if(rmq, ng=2, ng=1))))))); return(ng)} for(n=1, 
               100, k=ngoam3pf(n); if(k>=0, print1(k, ",")))
%Y A128644 Cf. A000001 (number of groups of order n), A037144 (numbers with at most 
               3 prime factors), A128604 (number of groups whose order divides p^6 
               for p a prime).
%Y A128644 Sequence in context: A128515 A119569 A066083 this_sequence A000001 A146002 
               A109087
%Y A128644 Adjacent sequences: A128641 A128642 A128643 this_sequence A128645 A128646 
               A128647
%K A128644 nonn
%O A128644 1,4
%A A128644 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Mar 20 2007