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A227501
Number of non-congruent solutions of x^2 - xy + y^2 == 1 (mod n).
1
1, 3, 6, 6, 6, 18, 6, 12, 18, 18, 12, 36, 12, 18, 36, 24, 18, 54, 18, 36, 36, 36, 24, 72, 30, 36, 54, 36, 30, 108, 30, 48, 72, 54, 36, 108, 36, 54, 72, 72, 42, 108, 42, 72, 108, 72, 48, 144, 42, 90, 108, 72, 54, 162, 72, 72, 108, 90, 60, 216, 60, 90, 108, 96
OFFSET
1,2
LINKS
FORMULA
Multiplicative: If p == 2 (mod 3) then a(p^s) = (p+1)*p^(s-1); if p == 1 (mod 3) then a(p^s) = (p-1)*p^(s-1); a(3^s) = 2*3^s.
From Amiram Eldar, Oct 13 2022: (Start)
a(n) = 2 * A227128(n) if n divisible by 3, and a(n) = A227128(n) otherwise.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 2/(3 * A086724) = 0.853276... . (End)
MATHEMATICA
Eisenstein[1] = 1; Eisenstein[n_] := Length@Select[Flatten[Table[{a, b}, {a, n}, {b, n}], 1], Mod[#[[1]]^2 + #[[2]]^2 - #[[1]]*#[[2]], n] == 1 &]; Array[Eisenstein, 100]
f[p_, e_] := If[Mod[p, 3] == 2, p + 1, p - 1]*p^(e - 1); f[3, e_] := 2*3^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 13 2022 *)
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 3, 2*3^f[i, 2], f[i, 1]^(f[i, 2] - 1) * (f[i, 1] + (-1)^(f[i, 1]%3))))}; \\ Amiram Eldar, Oct 13 2022
CROSSREFS
Sequence in context: A081289 A160504 A291801 * A294981 A079093 A153035
KEYWORD
nonn,mult
AUTHOR
STATUS
approved