1,1

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.

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

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.

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

Sequence in context: A044001 A008887 A096490 this_sequence A098136 A060793 A087004

Adjacent sequences: A056863 A056864 A056865 this_sequence A056867 A056868 A056869

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Sep 02 2000

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