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A088965 Number of solutions to x^2 + 2y^2 == 1 (mod n). 4
1, 2, 2, 4, 6, 4, 8, 16, 6, 12, 10, 8, 14, 16, 12, 32, 16, 12, 18, 24, 16, 20, 24, 32, 30, 28, 18, 32, 30, 24, 32, 64, 20, 32, 48, 24, 38, 36, 28, 96, 40, 32, 42, 40, 36, 48, 48, 64, 56, 60, 32, 56, 54, 36, 60, 128, 36, 60, 58, 48, 62, 64, 48, 128, 84, 40, 66, 64, 48, 96 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..10000

L. Toth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014) # 14.11.6.

FORMULA

Multiplicative with a(2^e) = 2^e for e <= 2, a(2^e) = 2^(e + 1) for e > 2, a(p^e) = (p-1)*p^(e-1) for p-2 mod 8 = +-1, a(p^e) = (p+1)*p^(e-1) for p-2 mod 8 = +-3. - Andrew Howroyd, Jul 13 2018

MAPLE

A088965 := proc(n) local a, x, y ; a := 0 ; for x from 0 to n-1 do for y from 0 to n-1 do if (x^2+2*y^2) mod n = 1 mod n then a := a+1 ; end if; end do; end do ; a ; end proc:

seq(A088965(n), n=1..70) ; # R. J. Mathar, Jan 07 2011

MATHEMATICA

a[1]=1; a[n_]:=Length@Rest@Union@Flatten@Table[If[Mod[i^2 + 2 j^2, n]==1, i+I j, 0], {i, 0, n-1}, {j, 0, n-1}]; Table[a[n], {n, 1, 80}] (* Vincenzo Librandi, Jul 16 2018 *)

PROG

(PARI) a(n)={my(v=vector(n)); for(i=0, n-1, v[i^2%n + 1]++); sum(i=0, n-1, v[i+1]*v[(1-2*i)%n + 1])} \\ Andrew Howroyd, Jul 09 2018

(PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 2^e*if(e>2, 2, 1), p^(e-1)*if(abs(p%8-2)==1, p-1, p+1)))} \\ Andrew Howroyd, Jul 09 2018

(MAGMA) [n eq 1 select 1 else #[x: x in [1..n], y in [1..n] | (x^2+2*y^2) mod n eq 1]: n in [1..80]]; // Vincenzo Librandi, Jul 16 2018

CROSSREFS

Cf. A087561, A060968.

Sequence in context: A318024 A320409 A096466 * A059474 A252828 A208314

Adjacent sequences:  A088962 A088963 A088964 * A088966 A088967 A088968

KEYWORD

mult,nonn

AUTHOR

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 28 2003

STATUS

approved

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Last modified December 9 08:41 EST 2021. Contains 349627 sequences. (Running on oeis4.)