

A227091


Number of solutions to x^2 == 1 (mod n) in Z[i]/nZ[i].


6



1, 2, 2, 4, 4, 4, 2, 8, 2, 8, 2, 8, 4, 4, 8, 8, 4, 4, 2, 16, 4, 4, 2, 16, 4, 8, 2, 8, 4, 16, 2, 8, 4, 8, 8, 8, 4, 4, 8, 32, 4, 8, 2, 8, 8, 4, 2, 16, 2, 8, 8, 16, 4, 4, 8, 16, 4, 8, 2, 32, 4, 4, 4, 8, 16, 8, 2, 16, 4, 16, 2, 16, 4, 8, 8, 8, 4, 16, 2, 32, 2, 8
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OFFSET

1,2


COMMENTS

Number of noncongruent solutions of x^2 + y^2 1 == 2xy == 0 (mod n).
This sequence combines A329586 (number of representative solutions of a^2  (b^2 + 1) == 0 (mod m) and 2*a*b == 0 (mod m) with a*b = 0), and those from A329589 (number of representative solutions of these two congruences but with a*b not 0).  Wolfdieter Lang, Dec 14 2019
In A226746 the positive n numbers with more than two representative solutions of the congruence z^2 = +1 (mod n) are given. This is therefore a proper subsequence of the present one.  Wolfdieter Lang, Dec 14 2019


LINKS

Eric M. Schmidt, Table of n, a(n) for n = 1..1000


FORMULA

Multiplicative with a(2^e) = 2^min(e, 3); a(p^e) = 4 for p == 1 (mod 4); a(p^e) = 2 for p == 3 (mod 4).  Eric M. Schmidt, Jul 09 2013


EXAMPLE

a(4)=4 because in Z[i]/4Z[i] the equation x^2==1 (mod 4) has 4 solutions: 1, 1+2i, 3 and 3+2i.


MAPLE

a:= n> mul(`if`(i[1]=2, 2^min(i[2], 3), `if`(
irem(i[1], 4)=1, 4, 2)), i=ifactors(n)[2]):
seq(a(n), n=1..100); # Alois P. Heinz, Feb 07 2020


MATHEMATICA

h[n_] := Flatten[Table[a + b I, {a, 0, n  1}, {b, 0, n  1}]]; a[1] = 1; a[n_] := Length@Select[h[n], Mod[#^2, n] == 1 &]; Table[a[n], {n, 2, 44}]


PROG

(Sage) def A227091(n) : return prod([4, 2^min(m, 3), 2][p%41] for (p, m) in factor(n)) # Eric M. Schmidt, Jul 09 2013
(PARI) a(n)=my(o=valuation(n, 2), f=factor(n>>o)[, 1]); prod(i=1, #f, if(f[i]%4==1, 4, 2))<<min(o, 3) \\ Charles R Greathouse IV, Dec 13 2013


CROSSREFS

Cf. A060594, A226746, A329586, A329589.
Sequence in context: A104202 A042946 A037202 * A165956 A263991 A065285
Adjacent sequences: A227088 A227089 A227090 * A227092 A227093 A227094


KEYWORD

nonn,mult


AUTHOR

José María Grau Ribas, Jun 30 2013


STATUS

approved



