

A139401


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.


3



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

1,2


COMMENTS

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. AdamsWatters, Jun 10 2011
Records of nonzero values in this sequence are in A066730.


LINKS



EXAMPLE

a(2) = 3 because there are no squares in the sequence 2, 5, 8, 11, 14, 17, 20, ...


PROG

(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


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



