A306271 -id:A306271 - cp's OEIS frontend

A306257 a(n) = t for the minimal integer k > t such that k^2 mod n = t^2 is a perfect square.

0, 0, 1, 0, 2, 2, 2, 1, 0, 3, 3, 2, 3, 3, 1, 0, 4, 0, 4, 4, 2, 4, 4, 1, 0, 5, 3, 5, 5, 2, 5, 2, 4, 5, 1, 0, 6, 6, 5, 3, 6, 4, 6, 6, 2, 6, 6, 1, 0, 0, 7, 7, 7, 6, 3, 5, 5, 7, 7, 2, 7, 7, 1, 0, 4, 8, 8, 8, 7, 2, 8, 3, 8, 8, 5, 8, 2, 7, 8, 1, 0, 9, 9, 4, 6, 9, 7, 9, 9, 4, 3, 9, 8, 9, 7, 2, 9, 0, 1, 0, 10, 8, 10, 9, 4

Offset: 1

Alois P. Heinz, Feb 13 2019

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000290, A077591, A306271 (values of k).

a:= proc(n) local k; for k from (s-> `if`(s^2

a(n) = 0 <=> n > 0 and n in { A000290 } union { A077591 }.

A306284 a(n) is the smallest positive integer x such that x > y >= 0 and n divides x^2 - y^2.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 3, 3, 6, 6, 4, 7, 8, 4, 4, 9, 6, 10, 6, 5, 12, 12, 5, 5, 14, 6, 8, 15, 8, 16, 6, 7, 18, 6, 6, 19, 20, 8, 7, 21, 10, 22, 12, 7, 24, 24, 7, 7, 10, 10, 14, 27, 12, 8, 9, 11, 30, 30, 8, 31, 32, 8, 8, 9, 14, 34, 18, 13, 12, 36, 9, 37, 38, 10, 20, 9, 16, 40, 9, 9, 42, 42, 10, 11, 44, 16

Offset: 1

Jianing Song, Feb 03 2019

Different from A306271: here x^2 mod n is not necessarily a square. For most n, a(n) != A306271(n).
It seems that n divides a(n)^2 if and only if n divides A306271(n)^2.
a(n) >= sqrt(n) with equality if and only if n is a square. - Robert Israel, Feb 05 2019

			a(10) = 6 because 10 divides 6^2 - 4^2 = 10, and 6 is the smallest possible value for x such that x > y >= 0 and that 10 divides x^2 - y^2.
a(87) = 16 because 87 divides 16^2 - 13^2 = 87, and 16 is the smallest possible value for x such that x > y >= 0 and that 87 divides x^2 - y^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A048152, A306271.

f:= proc(n) local S, x,t;
S:= {0}:
for x from 1 do
  t:= x^2 mod n;
  if member(t,S) then return x
    else S:= S union {t}
  fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 05 2019

a(n) = for(x=1, n, for(y=0, x-1, if((x^2-y^2)%n==0, return(x))))

from itertools import count
def A306284(n):
    y, a = 0, set()
    for x in count(0):
        if y in a: return x
        a.add(y)
        y = (y+(x<<1)+1)%n # Chai Wah Wu, Apr 25 2024

a(n^2) = n.
a(p) = (p + 1)/2 for primes p > 2.
For odd primes p and q, a(p*q) = (p+q)/2. - Robert Israel, Feb 08 2019

cp's OEIS Frontend

A306257 a(n) = t for the minimal integer k > t such that k^2 mod n = t^2 is a perfect square.

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A306284 a(n) is the smallest positive integer x such that x > y >= 0 and n divides x^2 - y^2.

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