A096336 - OEIS

%I #17 Mar 07 2017 04:04:21

%S 0,0,0,1,1,1,2,3,4,4,5,5,6,7,8,9,10,10,11,12,13,14,15,15,16,17,18,19,

%T 20,21,22,23,24,25,25,26,27,28,29,30,31,32,33,34,35,35,36,37,38,39,40,

%U 41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,55,56,57

%N Spin(2n+1) and Spin(2n+2) have torsion index 2^a(n).

%C First several terms agree with A169869 but the two sequences are distinct as can be seen where the values are 19 and 20. - _Skip Garibaldi_, Mar 05 2017

%H Michael De Vlieger, <a href="/A096336/b096336.txt">Table of n, a(n) for n = 0..10000</a>

%H Burt Totaro, <a href="https://www.math.ucla.edu/~totaro/papers/public_html/spin.pdf">The torsion index of the spin groups</a>, Duke Math. J. 129 (2005), no. 2, 249-290, <a href="http://doi.org/10.1215/S0012-7094-05-12923-4">doi:10.1215/S0012-7094-05-12923-4</a>.

%F a(n) is usually n-floor(log_2((n+1)n/2 + 1)), but is this number plus 1 if n = 2^e+b for nonnegative integers e, b such that 2b-a(b) <= e-3.

%t a[0] = 0; a[n_] := a[n] = Module[{e = Floor[Log2@n], b}, b = n - 2^e; n - Floor[Log2[(n + 1) n/2 + 1]] + Boole[2 b - a[b] <= e - 3]]; Table[a@ n, {n, 0, 120}] (* _Michael De Vlieger_, Mar 06 2017 *)

%o (Python)

%o import numpy as np

%o def a_typical(n):

%o     '''

%o     For most n, this is the value of a(n)

%o     '''

%o     return int(n - np.floor(np.log2( n*(n+1)/2 + 1)))

%o def a(n):

%o     '''

%o     The torsion index of Spin_{2n+1} and Spin_{2n+2} is 2^a(n)

%o     Totaro denotes it by u(ell)

%o     '''

%o     if n >= 0 and n <= 18:   # Table 1 in Totaro's paper

%o         return [0,0,0,1,1,1,2,3,4,4,5,5,6,7,8,9,10,10,11][n];

%o     maxe = int(np.floor(np.log2(n)))

%o     for e in range(maxe+1):

%o         b = n - 2**e

%o         if 2*b - a(b) <= e - 3: # occurs for n = 8, 16, 32, 33, ...

%o             return a_typical(n)+1

%o     return a_typical(n)

%o # _Skip Garibaldi_, Mar 05 2017

%K easy,nonn

%O 0,7

%A Richard Borcherds (reb(AT)math.berkeley.edu), Jun 28 2004

%E Edited and a(19)-a(49) added by _Skip Garibaldi_, Mar 05 2017

%E More terms from _Michael De Vlieger_, Mar 06 2017