|
| |
|
|
A056866
|
|
Orders of non-solvable groups, i.e. numbers which are not solvable numbers.
|
|
12
| |
|
|
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; 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 apart from the first trivial term of A060793 all the other numbers in these sequences 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.
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=1..2240 (orders < 10^5)
J. Pakianathan and K. Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.
R. Brauer, Investigation On Groups Of Even Order, I
R. Brauer, Investigation On Groups Of Even Order, II
W. Feit and J. G. Thompson, A Solvability Criterion For Finite Groups And Consequences
|
|
|
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.
|
|
|
CROSSREFS
| Cf. A003277, A051532, A056867, A056868, A001034.
Sequence in context: A182683 A174601 A096490 * A098136 A060793 A169823
Adjacent sequences: A056863 A056864 A056865 * A056867 A056868 A056869
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Sep 02 2000
|
|
|
EXTENSIONS
| More terms from Des MacHale (d.machale(AT)ucc.ie), Feb 19 2001
Further terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001
|
| |
|
|
