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
All terms are divisible by 4 and either 3 or 5. - Charles R Greathouse IV, Sep 11 2012
Subsequence of A056868 and hence of A060652. - Charles R Greathouse IV, Apr 16 2015, updated Sep 11 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) ~ k*n for some k (see, e.g., Erdős 1948). - Charles R Greathouse IV, Apr 17 2015
From Jianing Song, Apr 04 2022: (Start)
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
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2240 terms from T. D. 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.
Cindy Tsang, Qin Chao, On the solvability of regular subgroups in the holomorph of a finite solvable group, arXiv:1901.10636 [math.GR], 2019.
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
}; \\ Charles R Greathouse IV, Sep 11 2012
CROSSREFS
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