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Primitive non-solvable numbers: orders of non-solvable groups such that all groups with order a proper divisor of that order are solvable.
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%I #20 May 10 2018 03:31:32

%S 60,168,504,1092,2448,5616,6072,9828,25308,28224,32736,39732,51888,

%T 74412,150348,194472,285852,456288,546312,612468,721392,1024128,

%U 1285608,1934868,2097024,2165292,2328648,2588772,3594432,3822588,5544672,5848428,6324552,7174332,8487168,9095592

%N Primitive non-solvable numbers: orders of non-solvable groups such that all groups with order a proper divisor of that order are solvable.

%C Primitive elements of A056866; consequently, each term is divisible by 4 and either 3 or 5.

%C That is, numbers n such that n is in A056866, but no smaller m dividing n is in A056866. - _Charles R Greathouse IV_, May 09 2018

%H Charles R Greathouse IV, <a href="/A216480/b216480.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ kn^3 log^3 n, where k = 27/8. - _Charles R Greathouse IV_, Sep 11 2012

%o (PARI) list(lim)={

%o my(v=List([5616]),t);

%o forprime(p=2,log(lim)\log(8)+2,

%o listput(v,(4^p-1)<<p)

%o );

%o forprime(p=3,log(2*lim)\log(27)+2,

%o listput(v,3^p*(9^p\2))

%o );

%o forprime(p=3,log(lim)\log(32)+2,

%o listput(v,(4^p-1)*(2^p-1)<<(2*p))

%o );

%o forprime(p=7,sqrtn(2*lim,3)+1,

%o if(p%5>1 && p%5<4, listput(v,p^2\2*p))

%o );

%o vecsort(select(n->n<=lim,Vec(v)))

%o };

%Y Cf. A056866.

%K nonn,nice

%O 1,1

%A _Charles R Greathouse IV_, Sep 11 2012