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A128644
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Number of groups of order A037144(n).
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6
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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, 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, 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
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OFFSET
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1,4
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COMMENTS
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Number of groups whose order has at most 3 prime factors.
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.
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LINKS
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FORMULA
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EXAMPLE
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A037144(17) = 18 and there are 5 groups of order 18 (A000001(18) = 5), hence a(17) = 5.
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PROG
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(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 ] ];
(PARI) /* based on the formulas 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, ", ")))
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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