A136806 Nonsquares mod 65537.

3, 5, 6, 7, 10, 11, 12, 14, 20, 22, 23, 24, 27, 28, 29, 31, 39, 40, 41, 43, 44, 45, 46, 47, 48, 51, 54, 56, 57, 58, 59, 61, 62, 63, 65, 67, 73, 75, 78, 80, 82, 83, 85, 86, 88, 89, 90, 91, 92, 94, 95, 96, 99, 101, 102, 105, 108, 111, 112, 113, 114, 116, 118, 119

Offset

1,1

Comments

Because 65537 is a Fermat prime, these numbers are all primitive roots (mod 65537). Complement of A136805.

Links

Nathaniel Johnston, Table of n, a(n) for n = 1..32768 (full sequence)

OEIS Wiki, Index to sequences related to squares

Formula

a(n) + a(32769 - n) = 65537.

Example

Since 7 is not a perfect square, and there are no solutions to x^2 = 7 mod 65537, 7 is in the sequence.
Although 8 is not a perfect square either, there are solutions to x^2 = 8 mod 65537, such as x = 8160, so 8 is not in the sequence.

Maple

A136806 := {$(0..65536)}: for n from 0 to 65536 do A136806 := A136806 minus {n^2 mod 65537}: od: l:=sort(convert(A136806, list)): l[1..64]; # Nathaniel Johnston, Jun 23 2011
# Much more efficient:
S:= {$0..65536} minus {seq(i^2 mod 65537, i=0..65537/2)}:
A:= sort(convert(S, list)):
A[1..64]; # Robert Israel, Nov 15 2017

Mathematica

p = 65537; Select[Range[0, p - 1], JacobiSymbol[#, p] == -1 &]

Prog

(PARI) A136806=select( is_A136806(n)=!issquare(Mod(n, 65537)), [0..2^16]); \\ Strictly speaking, the is(.) function should include "&& n<65537" according to the intended meaning of the definition of this sequence. See A136804 for faster code, which would here cause a stack overflow for default settings. - M. F. Hasler, Nov 15 2017
(Scala) (1 to 65537).diff(((1: BigInt) to (65537: BigInt)).map(n => n * n % 65537)) // Alonso del Arte, Jan 17 2020

Crossrefs

Cf. A136805 (squares mod 65537); A136803 and A136804 ((non)squares mod 257).

Cf. A028730.

Sequence in context: A204232 A028730 A028747 * A253550 A295307 A073803

Adjacent sequences: A136803 A136804 A136805 * A136807 A136808 A136809

Keyword

fini,full,easy,nonn

Author

T. D. Noe, Jan 22 2008

Status

approved