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A096336
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Spin(2n+1) and Spin(2n+2) have torsion index 2^a(n).
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1
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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
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OFFSET
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0,7
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COMMENTS
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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
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LINKS
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FORMULA
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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.
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MATHEMATICA
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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 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Richard Borcherds (reb(AT)math.berkeley.edu), Jun 28 2004
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EXTENSIONS
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STATUS
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approved
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