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A139401
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If n is a square, a(n) is 0. Otherwise, a(n) is the smallest number k such that n is not a quadratic residue modulo k.
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3
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0, 3, 4, 0, 3, 4, 4, 3, 0, 4, 3, 5, 5, 3, 4, 0, 3, 4, 4, 3, 8, 4, 3, 7, 0, 3, 4, 5, 3, 4, 4, 3, 5, 4, 3, 0, 5, 3, 4, 7, 3, 4, 4, 3, 7, 4, 3, 5, 0, 3, 4, 5, 3, 4, 4, 3, 5, 4, 3, 9, 7, 3, 4, 0, 3, 4, 4, 3, 7, 4, 3, 5, 5, 3, 4, 7, 3, 4, 4, 3, 0, 4, 3, 9, 8, 3, 4, 5, 3, 4, 4, 3, 5, 4, 3, 7, 5, 3, 4, 0, 3, 4, 4, 3, 9
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OFFSET
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1,2
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COMMENTS
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I.e., if n is not a square, a(n) is the smallest number d for which a sequence that has a common difference of d contains n but that has no squares.
All nonzero values in this sequence are at least 3.
All nonzero values are prime powers, and every prime power except 2 appears in the sequence. This can be proved using the Chinese remainder theorem. - Franklin T. Adams-Watters, Jun 10 2011
Records of nonzero values in this sequence are in A066730.
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LINKS
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EXAMPLE
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a(2) = 3 because there are no squares in the sequence 2, 5, 8, 11, 14, 17, 20, ...
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PROG
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(Python)
import math
if int(math.sqrt(n)) == math.sqrt(n):
return 0
for pp in range(2, n + 2): # only really need to check prime powers
residues = frozenset(pow(k, 2, pp) for k in range(pp))
if n % pp not in residues:
(PARI) a(n) = if (issquare(n), 0, my(k=2); while (issquare(Mod(n, k)), k++); k); \\ Michel Marcus, Jun 25 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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