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
Examples
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.
Links
- Nathaniel Johnston, Table of n, a(n) for n = 1..32768 (full sequence)
- OEIS Wiki, Index to sequences related to squares
Programs
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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
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Mathematica
p = 65537; Select[Range[0, p - 1], JacobiSymbol[#, p] == -1 &]
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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
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Scala
(1 to 65537).diff(((1: BigInt) to (65537: BigInt)).map(n => n * n % 65537)) // Alonso del Arte, Jan 17 2020
Formula
a(n) + a(32769 - n) = 65537.
Comments