A028730 -id:A028730 - cp's OEIS frontend

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

T. D. Noe, Jan 22 2008

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

			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.

Nathaniel Johnston, Table of n, a(n) for n = 1..32768 (full sequence)
OEIS Wiki, Index to sequences related to squares

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

Cf. A028730.

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

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

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

(1 to 65537).diff(((1: BigInt) to (65537: BigInt)).map(n => n * n % 65537)) // Alonso del Arte, Jan 17 2020

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

A010379 Squares mod 17.

Original entry on oeis.org

0, 1, 2, 4, 8, 9, 13, 15, 16

Offset: 1

N. J. A. Sloane

Row 17 of A096008.

Cf. A028730, A136805.

Union[PowerMod[Range[17], 2, 17]] (* Alonso del Arte, Dec 17 2019 *)

[quadratic_residues(17)] # Zerinvary Lajos, May 24 2009

(1 to 17).map(n => (n * n) % 17).toSet.toSeq.sorted // Alonso del Arte, Dec 17 2019

A351026 Possible bases b > 17 which can be used in Pepin's test to check the primality of any Fermat number greater than 5 only in the case when the base b is smaller than the tested number.

Original entry on oeis.org

51, 85, 102, 119, 170, 204, 238, 291, 340, 408, 459, 476, 485, 579, 582, 663, 679, 680, 697, 723, 765, 771, 816, 867, 918, 952, 965, 970, 1071, 1105, 1158, 1164, 1205, 1275, 1285, 1326, 1351, 1358, 1360, 1394, 1445, 1446, 1530, 1542, 1547, 1632, 1687, 1734, 1785

Offset: 1

Arkadiusz Wesolowski, Jan 29 2022

nonn

R. D. Carmichael, Fermat numbers F(n) = 2^(2^n) + 1, Amer. J. Math., 26 (1919), 137-146.
Eric Weisstein's World of Mathematics, Pepin's Test

Cf. A028730, A129802, A136804, A136806.

for(b=18, 1785, a=q=0; until(b-2<16^(2^a), a++; if(!(kronecker(b, 16^(2^(a-1))+1)==-1), q=1; break)); if(q==1, k=b/2^valuation(b, 2); if(k>1, i=logint(k, 2); m=Mod(2, k); z=znorder(m); e=znorder(Mod(2, z/2^valuation(z, 2))); t=0; for(c=1, e, if(kronecker(lift(m^2^(i+c))+1, k)==-1, t++, break)); if(t==e, print1(b, ", ")))));

A positive integer b belongs to this sequence if and only if the Jacobi symbol J(b,F(m)) has value 0 or 1 for some 5 < F(m) < b, and J(b,F(m)) = 1 only for a finite number of Fermat numbers F(m) = 2^(2^m) + 1.

A242860 a(n) is the least k >= 2 such that k is neither a square modulo n nor a primitive root (mod n), or 0 if no such value exists.

Original entry on oeis.org

0, 2, 0, 2, 6, 2, 3, 2, 10, 2, 5, 6, 2, 2, 0, 2, 8, 2, 2, 2, 22, 2, 5, 2, 3, 2, 12, 2, 6, 2, 2, 6, 2, 2, 6, 2, 2, 2, 3, 2, 2, 2, 2, 10, 46, 2, 6, 2, 2, 2, 23, 2, 2, 2, 2, 2, 58, 2, 8, 6, 2, 2, 2, 2, 3, 2, 2, 2, 14, 2, 7, 2, 2, 2, 2, 2, 12, 2, 3, 3, 82, 2, 2, 2

Offset: 3

Arkadiusz Wesolowski, May 24 2014

nonn

a(A019434(n)) = 0 for n >= 1.

Eric Weisstein's World of Mathematics, Primitive Root

Cf. A028730, A136804, A136806.

lst:=[]; for n in [3..86] do v:=0; for r in [2..n-1] do if not IsSquare(ResidueClassRing(n)! r) and not IsPrimitive(r, n) then v:=r; break; end if; end for; lst:=Append(lst, v); end for; lst;

cp's OEIS Frontend

A136806 Nonsquares mod 65537.

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A010379 Squares mod 17.

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A351026 Possible bases b > 17 which can be used in Pepin's test to check the primality of any Fermat number greater than 5 only in the case when the base b is smaller than the tested number.

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A242860 a(n) is the least k >= 2 such that k is neither a square modulo n nor a primitive root (mod n), or 0 if no such value exists.

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