A186772 -id:A186772

The OEIS is supported by the many generous donors to the OEIS Foundation.

Search: a186772 -id:a186772

Displaying 1-3 of 3 results found.

page 1

A124901

Smallest order of any nonsolvable transitive Galois group for a polynomial of degree n.

+10
4

60, 60, 168, 168, 504, 60, 660, 60, 5616, 168, 60, 336, 4080, 180, 60822550204416000, 60, 168, 1320, 10200960, 120, 300, 5616, 1512, 168, 4420880996869850977271808000000, 60

(list; graph; refs; listen; history; text; internal format)

OFFSET

5,1

COMMENTS

These transitive groups are in MAGMA classification respectively:

a(5)=5T4, a(6)=6T12, a(7)=7T5, a(8)=8T37, a(9)=9T27,

a(10)=10T6, a(11)=11T5, a(12)=12T33, a(13)=13T7, a(14)=14T10,

a(15)=15T5, a(16)=16T713, a(17)=17T6, a(18)=18T90, a(19)=19T7,

a(20)=20T15, a(21)=21T14, a(22)=22T13, a(23)=23T4,

a(24)=24T201, a(25)=25T29, a(26)=26T39, a(27)=27T390,

a(28)=28T32, a(29)=28T7, a(30)=30T9.

LINKS

Table of n, a(n) for n=5..30.

EXAMPLE

a(8)=336 because nonsolvable Galois group PGL(2,7)=L(8) has order 336.

CROSSREFS

Cf. A124900, A186772.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Nov 12 2006

EXTENSIONS

a(4) corrected and a(11)-a(30) by Artur Jasinski, Feb 26 2011

STATUS

approved

A124900

Largest order of any solvable transitive Galois group for an irreducible polynomial of degree n.

+10
3

1, 2, 6, 24, 20, 72, 42, 1152, 1296, 800, 110, 82944, 156, 3528, 155520, 7962624, 272, 2239488, 342, 159252480, 11757312, 225280, 506, 13759414272, 64000000, 1277952, 13060694016, 192631799808, 812, 48372940800

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

These transitive groups are in classification of MAGMA:

a(1)=1T1,a(2)=2T1,a(3)=3T2,a(4)=4T5,a(5)=5T3,a(6)=6T13,

a(7)=7T4,a(8)=8T47,a(9)=9T31,a(10)=10T33,a(11)=11T4,

a(12)=12T294,a(13)=13T6,a(14)=14T45,a(15)=15T87,

a(16)=16T1947,a(17)=17T5,a(18)=18T945,a(19)=19T6,

a(20)=20T1067,a(21)=21T142,a(22)=22T37,a(23)=23T5,

a(24)=24T24921,a(25)=25T179,a(26)=26T79,a(27)=27T2372,

a(28)=28T1773,a(29)=29T6,a(30)=30T5358.

Conjecture: The sequence a(prime(n)), which begins 2, 6, 20, 42, 110, 156, 272, 342, 506, 812, increases without bound. It appears that a(prime(n)) may equal prime(n)(prime(n)-1), which is A036689. - Artur Jasinski, Feb 26 2011

LINKS

Table of n, a(n) for n=1..30.

EXAMPLE

a(9)=1296 because solvable Galois group T9_31 (in MAGMA's list) has order 1296

CROSSREFS

Cf. A099732, A124901, A186772.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Nov 12 2006

EXTENSIONS

a(11)-a(30) from Artur Jasinski, Feb 26 2011

STATUS

approved

A186860

Largest coefficient of (1)(1+2x)(1+2x+3x^2)*...*(1+2x+3x^2+...+(n+1)*x^n).

+10
2

1, 2, 7, 49, 562, 9132, 207915, 6296448, 239972192, 11427298486, 661227186254, 45688884832738, 3716852205228166, 351101915633367990, 38275029480566516322, 4750162039324230600200, 666311679640315952033655, 105085327413072323807645048

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..200

FORMULA

Conjecture: a(n) ~ 3^(3/2) * sqrt(Pi) * n^(2*n + 1/2) / (2^(n-1) * exp(2*n)). - Vaclav Kotesovec, Jan 05 2023

MATHEMATICA

f[n_] := Max@ CoefficientList[ Expand@ Product[ Sum[(i + 1)*x^i, {i, 0, j}], {j, n - 1}], x]; Array[f, 18]

PROG

(Sage)

def A186860(n):

p = prod(sum(i*x^(i-1) for i in (1..k)) for k in (1..n))

return Integer(max(p.coefficients())[0]) # D. S. McNeil, Feb 28 2011

CROSSREFS

Cf. A000140, A186772.

KEYWORD

easy,nonn

AUTHOR

Robert G. Wilson v, Feb 27 2011

STATUS

approved

page 1

Search completed in 0.007 seconds