A034268 -id:A034268 - cp's OEIS frontend

A019320 Cyclotomic polynomials at x=2.

2, 1, 3, 7, 5, 31, 3, 127, 17, 73, 11, 2047, 13, 8191, 43, 151, 257, 131071, 57, 524287, 205, 2359, 683, 8388607, 241, 1082401, 2731, 262657, 3277, 536870911, 331, 2147483647, 65537, 599479, 43691, 8727391, 4033, 137438953471, 174763, 9588151, 61681

Offset: 0

Simon Plouffe

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000
Joerg Arndt, Matters Computational (The Fxtbook)
Index entries for cyclotomic polynomials, values at X

a(n) = A063696(n) - A063698(n) for up to n=104.
Same sequence in binary: A063672.
Cf. A054432, A020501, A034268.

with(numtheory,cyclotomic); f := n->subs(x=2,cyclotomic(n,x)); seq(f(i),i=0..64);

Join[{2}, Table[Cyclotomic[n, 2], {n, 1, 40}]] (* Jean-François Alcover, Jun 14 2013 *)

vector(20,n,polcyclo(n,2)) \\ Charles R Greathouse IV, May 18 2011

(lcm_{k=1..n} (2^k - 1))/lcm_{k=1..n-1} (2^k - 1), n > 1. - Vladeta Jovovic, Jan 20 2002
Let b(1) = 1 and b(n+1) = lcm(b(n), 2^n-1) then Phi(n,2) = b(n+1)/b(n) = a(n). - Thomas Ordowski, May 08 2013
a(0) = 2; for n > 0, a(n) = (2^n-1)/gcd(a(0)*a(1)*...*a(n-1), 2^n-1). - Thomas Ordowski, May 11 2013

A066845 a(n) = (lcm_{k=0..n} (2^k + 1))/(lcm_{k=0..n-1} (2^k + 1)).

Original entry on oeis.org

3, 5, 3, 17, 11, 13, 43, 257, 57, 205, 683, 241, 2731, 3277, 331, 65537, 43691, 4033, 174763, 61681, 5419, 838861, 2796203, 65281, 1016801, 13421773, 261633, 15790321, 178956971, 80581, 715827883, 4294967297, 1397419, 3435973837

Offset: 1

Vladeta Jovovic, Jan 20 2002

The primitive part of 2^n + 1. Bisection of A019320. - T. D. Noe, Jul 24 2008

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A051844, A034268, A019320.

Cf. A019320.

Cyclotomic[2*Range[40],2] (* Harvey P. Dale, Apr 19 2019 *)

a(n) = polcyclo(2*n, 2); \\ Michel Marcus, Mar 06 2015

a(n) = cyclotomic(2*n, 2). - Vladeta Jovovic, Apr 05 2004

A051844 a(n) = LCM_{k=0..n} (2^k + 1).

Original entry on oeis.org

2, 6, 30, 90, 1530, 16830, 218790, 9407970, 2417848290, 137817352530, 28252557268650, 19296496614487950, 4650455684091595950, 12700394473254148539450, 41619192688853844763777650, 13775952780010622616810402150, 902834617343556174437903325704550

Offset: 0

Jeffrey Shallit, Apr 20 2000

nonn

			a(3) = lcm(2, 3, 5) = 30.

Cf. A034268.

Cf. A019320.

Module[{nn=20,c},c=Table[2^n+1,{n,0,nn}];Table[LCM@@Take[c,n],{n,nn}]] (* Harvey P. Dale, Aug 04 2017 *)

a(n) = {ret = 1; for (k=0, n, ret = lcm(ret, 2^k+1)); return(ret);} \\ Michel Marcus, May 24 2013

from math import lcm
from itertools import accumulate
def aupton(nn): return list(accumulate((2**k+1 for k in range(nn+1)), lcm))
print(aupton(16)) # Michael S. Branicky, Jul 04 2022

a(n) = lcm(2, 3, 5, ..., 2^n + 1).

Product_{k=1..n} cyclotomic(2*k-2, 2). - Vladeta Jovovic, Apr 05 2004

More terms from Harvey P. Dale, Aug 04 2017

A211171 Exponent of general linear group GL(n,2).

Original entry on oeis.org

1, 6, 84, 420, 26040, 78120, 9921240, 168661080, 24624517680, 270869694480, 554470264600560, 7208113439807280, 59041657185461430480, 2538791258974841510640, 383357480105201068106640, 98522872387036674503406480, 25826982813282567927671981480160

Offset: 1

Alexander Gruber, Jan 31 2013

nonn

a(n) is the smallest integer for which x^a(n) = 1 for any x in GL(n,2).

			n = 2: GL(2,2) is isomorphic to S3 which has exponent 6 (see: A003418).
n = 3: The set of element orders of GL(3,2) is {1,2,3,4,7} so the exponent is 84.
n = 5: The set of element orders of GL(5,2) is {1,2,3,4,5, 6,7,8,12,14, 15,21,31} so the exponent is 26040 (see: A053651).

Alexander Gruber, Table of n, a(n) for n = 1..100
Gert Almkvist, Powers of a matrix with coefficients in a Boolean ring, Proc. Amer. Math. Soc. 53 (1975), 27-31. See u_n.
Eugene Karolinsky and Dmytro Seliutin, Carmichael numbers for GL(m), arXiv:2001.10315 [math.NT], 2020; where a(n) is noted as K2(n), see page 1.
MathStackExchange, Exponent of GL(n,q)

Cf. A003418, A053651.
Cf. A006951 (number of conjugacy classes in GL(n,2)).
Cf. A034268, A062383.

for n in [1..18] do
Exponent(GL(n,2));
end for;

with(numtheory):
a:= proc(n) local t; t:= 2^ilog2(n);
      `if`(tAlois P. Heinz, Feb 04 2013

f[q_, n_] := With[{p = Sort[Divisors[q]][[2]]},
  p^Ceiling[Log[p, n]] Product[Cyclotomic[k, q], {k, n}]]; f[2,#]&/@Range[100]

a(n) = 2^ceil(log(n)/log(2))*prod(k=1, n, polcyclo(k, 2)); \\ Michel Marcus, Jan 29 2020

a(n) = 2^ceiling(log_2(n)) * Product_{k=1..n} (k-th cyclotomic polynomial evaluated at 2).

a(n) = A034268(n)*A062383(n+1). - Michel Marcus, Jul 29 2022

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A019320 Cyclotomic polynomials at x=2.

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A066845 a(n) = (lcm_{k=0..n} (2^k + 1))/(lcm_{k=0..n-1} (2^k + 1)).

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A051844 a(n) = LCM_{k=0..n} (2^k + 1).

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A211171 Exponent of general linear group GL(n,2).

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