A064487 Order of twisted Suzuki group Sz(2^(2*n + 1)), also known as the group 2B2(2^(2*n + 1)).

20, 29120, 32537600, 34093383680, 35115786567680, 36011213418659840, 36888985097480437760, 37777778976635853209600, 38685331082014736871587840, 39614005699412557795646504960, 40564799864499450381466515537920

Offset: 0

Author

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 15 2001

Keywords

• nonn
• easy

Comments

Every term in A056866 is divisible by 12 or 20. Those terms that are not divisible by 12 are divisible by a term from this sequence. - Charles R Greathouse IV via Danny Rorabaugh, Apr 21 2015

For n >= 3, a(n) has at least 5 distinct prime factors. See my link for a proof. - Jianing Song, Apr 04 2022

References

• R. W. Carter, Simple Groups of Lie Type, Wiley 1972, Chap. 14.
• J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi. See ATLAS v. 3
• Jianing Song, Proof that a(n) has at least 5 distinct prime factors for n >= 3
• Michio Suzuki, A new type of simple groups of finite order, Proc Natl Acad Sci U S A. 46:6 (1960), pp. 868-870.
• Index entries for linear recurrences with constant coefficients, signature (1360,-365568,22282240,-268435456).

Crossrefs

Cf. A037250, A064583. A257391 is a subsequence.

Programs

GAP

g := Sz(32); s := Size(g);

Magma

[ #Sz(2^(2*n+1)) : n in [0..10]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

Mathematica

LinearRecurrence[{1360,-365568,22282240,-268435456},{20,29120,32537600,34093383680},20] (* Harvey P. Dale, Sep 08 2018 *)

PARI

a(n)=my(t=2^(2*n+1)); t^2*(t-1)*(t^2+1) \\ Charles R Greathouse IV, Apr 21 2015

PARI

Vec(20*(1+128*x)*(1-32*x+16384*x^2)/((1-16*x)*(1-64*x)*(1-256*x)*(1-1024*x)) + O(x^20)) \\ Colin Barker, Dec 25 2015

Formula

a(n) = q^4*(q^2-1)*(q^4+1), where q^2 = 2^(2*n+1).

G.f.: 20*(1+128*x)*(1-32*x+16384*x^2) / ((1-16*x)*(1-64*x)*(1-256*x)*(1-1024*x)). - Colin Barker, Dec 25 2015