|
|
A056866
|
|
Orders of non-solvable 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
(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 three sequences A001034, A060793, A056866: 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 [Corrected by Isaac Saffold, Aug 09 2021]
Insoluble group orders can be derived from A001034 (simple non-cyclic orders): k is an insoluble order iff k is a multiple of a simple non-cyclic 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, August-September 2000, 631-634.
|
|
FORMULA
|
A positive integer k is a non-solvable 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^2-1)/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^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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Further terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001
|
|
STATUS
|
approved
|
|
|
|