A096336 Spin(2n+1) and Spin(2n+2) have torsion index 2^a(n).

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, 20, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 55, 56, 57

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

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

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Burt Totaro, The torsion index of the spin groups, Duke Math. J. 129 (2005), no. 2, 249-290, doi:10.1215/S0012-7094-05-12923-4.

Programs

Mathematica

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 *)

Python

import numpy as np
def a_typical(n):
    '''
    For most n, this is the value of a(n)
    '''
    return int(n - np.floor(np.log2( n*(n+1)/2 + 1)))
def a(n):
    '''
    The torsion index of Spin_{2n+1} and Spin_{2n+2} is 2^a(n)
    Totaro denotes it by u(ell)
    '''
    if n >= 0 and n <= 18:   # Table 1 in Totaro's paper
        return [0,0,0,1,1,1,2,3,4,4,5,5,6,7,8,9,10,10,11][n];
    maxe = int(np.floor(np.log2(n)))
    for e in range(maxe+1):
        b = n - 2**e
        if 2*b - a(b) <= e - 3: # occurs for n = 8, 16, 32, 33, ...
            return a_typical(n)+1
    return a_typical(n)
# Skip Garibaldi, Mar 05 2017

Formula

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.

Extensions

Edited and a(19)-a(49) added by Skip Garibaldi, Mar 05 2017

More terms from Michael De Vlieger, Mar 06 2017