A087780 Number of non-congruent solutions to x^2 == 2 mod n.

1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0

Offset

1,7

Links

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^m) = 2 for p == 1, 7 (mod 8); a(p^m) = 0 for p == 3, 5 (mod 8); a(2^1) = 1; a(2^m) = 0 for m > 1. - Eric M. Schmidt, Apr 20 2013

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(sqrt(2)+1)/(sqrt(2)*zeta(2)) = A196525/A013661 = 0.37887551404073012021... . - Amiram Eldar, Nov 21 2023

Mathematica

f[2, e_] := Boole[e == 1]; f[p_, e_] := If[MemberQ[{1, 7}, Mod[p, 8]], 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

Prog

(Sage)
def A087780(n) :
    res = 1
    for (p, m) in factor(n) :
        if p % 8 in [1, 7] : res *= 2
        elif not (p==2 and m==1) : return 0
    return res
# Eric M. Schmidt, Apr 20 2013
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, (f[i, 2] == 1), if(f[i, 1]%8 == 1 || f[i, 1]%8 == 7, 2, 0))); } \\ Amiram Eldar, Nov 21 2023

Crossrefs

Cf. A038873, A057126, A060594.

Cf. A013661, A196525.

Sequence in context: A332006 A086016 A103270 * A082523 A186034 A280843

Adjacent sequences: A087777 A087778 A087779 * A087781 A087782 A087783

Keyword

mult,nonn,easy

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Extensions

More terms from David Wasserman, Jun 17 2005

Status

approved