cp's OEIS Frontend

A086933 Number of solutions to x^2 + y^2 = 0 mod n.

%I A086933 #37 Mar 24 2019 14:39:23
%S A086933 1,2,1,4,9,2,1,8,9,18,1,4,25,2,9,16,33,18,1,36,1,2,1,8,65,50,9,4,57,
%T A086933 18,1,32,1,66,9,36,73,2,25,72,81,2,1,4,81,2,1,16,49,130,33,100,105,18,
%U A086933 9,8,1,114,1,36,121,2,9,64,225,2,1,132,1,18,1,72,145,146,65,4,1,50,1,144,81
%N A086933 Number of solutions to x^2 + y^2 = 0 mod n.
%C A086933 Sum_{n<N} a(n) ~ (Pi/(8*G))*N^2 as N approaches infinity, where G is Catalan's constant. - _Steven Finch_, Feb 05 2007
%H A086933 Andrew Howroyd, <a href="/A086933/b086933.txt">Table of n, a(n) for n = 1..10000</a>
%H A086933 S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Series involving arithmetric functions</a>.
%H A086933 N. Gafurov, <a href="http://mi.mathnet.ru/eng/tm1417">On the number of divisors of a quadratic form</a>, Proc. Steklov Inst. Math. 200 (1993) 137-148.
%H A086933 L. Tóth, <a href="http://arxiv.org/abs/1404.4214">Counting solutions of quadratic congruences in several variables revisited</a>, arXiv preprint arXiv:1404.4214 [math.NT], 2014.
%H A086933 L. Toth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Toth/toth12.html">Counting Solutions of Quadratic Congruences in Several Variables Revisited</a>, J. Int. Seq. 17 (2014) # 14.11.6.
%H A086933 G. Yu, <a href="http://dx.doi.org/10.4153/CMB-2000-032-3">On the number of divisors of the quadratic form m^2+n^2</a>, Canad. Math. Bull. 43 (2000) 239-256.
%F A086933 Multiplicative with a(2^e)=2^e, a(p^e)=p^(e-(e mod 2)) if p mod 4=3, a(p^e)=((p-1)*e+p)*p^(e-1) if p mod 4<>3 and p<>2. - _Vladeta Jovovic_, Sep 22 2003
%F A086933 From _Peter Bala_, Mar 24 2019: (Start)
%F A086933 a(n) = n*Sum_{d|n, d odd} (-1)^((d-1)/2)*phi(d)/d.
%F A086933 O.g.f.: Sum_{n odd} (-1)^((n-1)/2)*phi(n)*x^n/(1 - x^n)^2. (End)
%t A086933 a[n_] := a[n] = Module[{f, p, e}, f = FactorInteger[n]; Switch[f, {{2, _}}, Return[n], {{_, _}}, {p, e} = f[[1]]; If[Mod[p, 4] == 3, Return[p^(e - Mod[e, 2])], Return[((p-1)*e+p)*p^(e-1)]], _, Times @@ (a[#[[1]]^#[[2]]]& /@ f)]];
%t A086933 Array[a, 81] (* _Jean-François Alcover_, Aug 21 2018, after _Vladeta Jovovic_ *)
%o A086933 (PARI) ap(p,e)=if(p%4<2, ((p-1)*e+p)*p^(e-1), p^(e - e%2))
%o A086933 a(n)=my(o=valuation(n,2),f=factor(n>>o)); prod(i=1,#f~, ap(f[i,1],f[i,2]))<<o \\ _Charles R Greathouse IV_, Dec 06 2016
%Y A086933 Cf. A062803.
%K A086933 mult,nonn
%O A086933 1,2
%A A086933 Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 21 2003
%E A086933 More terms from _John W. Layman_, Sep 22 2003