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

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

1,2

Comments

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

Links

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

Formula

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

Example

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.

Maple

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

Prog

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

Crossrefs

Cf. A048152, A306271.

Sequence in context: A065458 A341121 A144000 * A318816 A085202 A096009

Adjacent sequences: A306281 A306282 A306283 * A306285 A306286 A306287

Keyword

nonn,look

Author

Jianing Song, Feb 03 2019

Status

approved