cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A124901 Smallest order of any nonsolvable transitive Galois group for a polynomial of degree n.

Original entry on oeis.org

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
Offset: 5

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Author

Artur Jasinski, Nov 12 2006

Keywords

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.

Examples

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

Crossrefs

Cf. A124900, A186772.

Extensions

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

A124900 Largest order of any solvable transitive Galois group for an irreducible polynomial of degree n.

Original entry on oeis.org

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
Offset: 1

Views

Author

Artur Jasinski, Nov 12 2006

Keywords

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

Examples

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

Crossrefs

Cf. A099732, A124901, A186772.

Extensions

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

A186860 Largest coefficient of (1)(1+2x)(1+2x+3x^2)*...*(1+2x+3x^2+...+(n+1)*x^n).

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Feb 27 2011

Keywords

Links

Crossrefs

Cf. A000140, A186772.

Programs

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

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
Showing 1-3 of 3 results.

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