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A056866 Orders of non-solvable groups, i.e. numbers which are not solvable numbers. 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A number is solvable if every group of that order is solvable.

This comment is about the 4 sequences A001034, A060793, A056866, A056868: The Feit Thompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001

Insoluble group orders can be derived from A001034 (simple non-cyclic orders): n is an insoluble order iff n is a multiple of a simple non-cyclic order. - Des MacHale.

All terms are divisible by 4 and either 3 or 5. - Charles R Greathouse IV, Sep 11 2012

Subsequence of A060652. - Charles R Greathouse IV, Apr 16 2015

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) ~ kn for some k (see, e.g., Erdős 1948). - Charles R Greathouse IV, Apr 17 2015

LINKS

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2240 terms from Noe)

R. Brauer, Investigation On Groups Of Even Order, I

R. Brauer, Investigation On Groups Of Even Order, II

P. Erdős, On the density of some sequences of integers, Bull. Amer. Math. Soc. 54 (1948), pp. 685-692. See p. 685.

W. Feit and J. G. Thompson, A Solvability Criterion For Finite Groups And Consequences, Proc. N. A. S. 48 (6) (1962) 968

J. Pakianathan and K. Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.

FORMULA

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

}; \\ Charles R Greathouse IV, Sep 11 2012

CROSSREFS

Cf. A003277, A051532, A056867, A056868, A001034, A060652.

Sequence in context: A252961 A252962 A096490 * A098136 A060793 A169823

Adjacent sequences:  A056863 A056864 A056865 * A056867 A056868 A056869

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Sep 02 2000

EXTENSIONS

More terms from Des MacHale, Feb 19 2001

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

STATUS

approved

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Last modified July 2 05:46 EDT 2015. Contains 259128 sequences.