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A372494
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For n >= 1, a(n) is the least k >= 0 such that k^2 + k + 1 is divisible by 2^n - 1 or a(n) = -1 if no such k exists.
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1
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0, 1, 2, -1, 5, -1, 19, -1, 81, -1, -1, -1, 90, 1885, 4799, -1, 21905, -1, 69492, -1, 99206, -1, -1, -1, 585398, 8985436, 32569980, -1, -1, -1, 634005911, -1, -1, -1, -1, -1, 7348910325, 40154162248, 28169995830, -1, -1, -1, -1, -1, 749259285289, -1, -1, -1, 22640169503650
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OFFSET
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1,3
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LINKS
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EXAMPLE
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n = 5: (k^2 + k + 1) / (2^5 - 1) is true for the least k = 5, thus a(5) = 5.
n = 7: (k^2 + k + 1) / (2^7 - 1) is true for the least k = 19, thus a(7) = 19.
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PROG
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(Python)
from sympy import sqrt_mod_iter
k = (1<<n)-1
k2 = k>>1
return min((d>>1 if d&1 else (d>>1)+k2 for d in sqrt_mod_iter(-3, k)), default=-1) # Chai Wah Wu, May 03 2024
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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