

A056866


Orders of nonsolvable groups, i.e., numbers that are not solvable numbers.


18



60, 120, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1020, 1080, 1092, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500, 1512, 1560, 1620, 1680, 1740, 1800, 1848, 1860, 1920, 1980, 2016, 2040
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OFFSET

1,1


COMMENTS

A number is solvable if every group of that order is solvable.
This comment is about the three sequences A001034, A060793, A056866: The FeitThompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even.  Ahmed Fares (ahmedfares(AT)mydeja.com), May 08 2001 [Corrected by Isaac Saffold, Aug 09 2021]
Insoluble group orders can be derived from A001034 (simple noncyclic orders): k is an insoluble order iff k is a multiple of a simple noncyclic order.  Des MacHale
The primitive elements are A257146. Since the sum of the reciprocals of the terms of that sequence converges, this sequence has a natural density and so a(n) ~ k*n for some k (see, e.g., Erdős 1948).  Charles R Greathouse IV, Apr 17 2015
Burnside's p^a*q^b theorem says that a finite group whose order has at most 2 distinct prime factors is solvable, hence all terms have at least 3 distinct prime factors.
Terms not divisible by 12 are divisible by 320 and have at least 4 distinct prime factors (cf. A257391). (End)


LINKS

J. Pakianathan and K. Shankar, Nilpotent numbers, Amer. Math. Monthly, 107, AugustSeptember 2000, 631634.


FORMULA

A positive integer k is a nonsolvable number if and only if it is a multiple of any of the following numbers: a) 2^p*(2^(2*p)1), p any prime. b) 3^p*(3^(2*p)1)/2, p odd prime. c) p*(p^21)/2, p prime greater than 3 such that p^2 + 1 == 0 (mod 5). d) 2^4*3^3*13. e) 2^(2*p)*(2^(2*p)+1)*(2^p1), p odd prime.


MATHEMATICA

ma[n_] := For[k = 1, True, k++, p = Prime[k]; m = 2^p*(2^(2*p)  1); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; mb[n_] := For[k = 2, True, k++, p = Prime[k]; m = 3^p*((3^(2*p)  1)/2); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; mc[n_] := For[k = 3, True, k++, p = Prime[k]; m = p*((p^2  1)/2); If[Mod[p^2 + 1, 5] == 0, If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]]; md[n_] := Mod[n, 2^4*3^3*13] == 0; me[n_] := For[k = 2, True, k++, p = Prime[k]; m = 2^(2*p)*(2^(2*p) + 1)*(2^p  1); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; notSolvableQ[n_] := OddQ[n]  ma[n]  mb[n]  mc[n]  md[n]  me[n]; Select[ Range[3000], notSolvableQ] (* JeanFrançois Alcover, Jun 14 2012, from formula *)


PROG

(PARI) is(n)={
if(n%5616==0, return(1));
forprime(p=2, valuation(n, 2),
if(n%(4^p1)==0, return(1))
);
forprime(p=3, valuation(n, 3),
if(n%(9^p\2)==0, return(1))
);
forprime(p=3, valuation(n, 2)\2,
if(n%((4^p+1)*(2^p1))==0, return(1))
);
my(f=factor(n)[, 1]);
for(i=1, #f,
if(f[i]>3 && f[i]%5>1 && f[i]%5<4 && n%(f[i]^2\2)==0, return(1))
);
0


CROSSREFS



KEYWORD

nonn,easy,nice


AUTHOR



EXTENSIONS

Further terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001


STATUS

approved



