cp's OEIS Frontend

A208895 Number of non-congruent solutions to x^2 + y^2 + z^2 + t^2 == 1 (mod n).

%I A208895 #40 Oct 18 2022 07:27:13
%S A208895 1,8,24,64,120,192,336,512,648,960,1320,1536,2184,2688,2880,4096,4896,
%T A208895 5184,6840,7680,8064,10560,12144,12288,15000,17472,17496,21504,24360,
%U A208895 23040,29760,32768,31680,39168,40320,41472,50616,54720,52416,61440,68880,64512
%N A208895 Number of non-congruent solutions to x^2 + y^2 + z^2 + t^2 == 1 (mod n).
%H A208895 Amiram Eldar, <a href="/A208895/b208895.txt">Table of n, a(n) for n = 1..10000</a>
%H A208895 László 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 A208895 László 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), Article 14.11.6.
%F A208895 Conjecture: a(n) = n*Sum_{d|2*n} d^2*mu(2*n/d)/3. - _Gionata Neri_, Feb 18 2018
%F A208895 From _Amiram Eldar_, Oct 18 2022: (Start)
%F A208895 Multiplicative with a(p^e) = p^(3*e)*(1-1/p^2) if p > 2, and a(2^e) = 8^e.
%F A208895 Sum_{k=1..n} a(k) ~ c * n^4 + O(n^3), where c = 2/(7*zeta(3)) = 0.237687... (Tóth, 2014). (End)
%p A208895 A208895 := proc(n)
%p A208895     local a,pe,p,nu ;
%p A208895     a := 1 ;
%p A208895     for pe in ifactors(n)[2] do
%p A208895         p := op(1,pe) ;
%p A208895         nu := op(2,pe) ;
%p A208895         if p > 2 then
%p A208895             a := a*p^(3*nu)*(1-1/p^2) ;
%p A208895         else
%p A208895             a := a*8^nu ;
%p A208895         end if;
%p A208895     end do:
%p A208895     a ;
%p A208895 end proc:
%p A208895 seq(A208895(n),n=1..20) ; # _R. J. Mathar_, Jun 23 2018
%t A208895 a[n_] := Length[Union[Flatten[Table[If[Mod[x^2 + y^2 + z^2 + t^2, n] == 1, {x, y, z, t}], {x, n}, {y, n}, {z, n}, {t, n}], 3]]] - 1; Join[{1}, Table[a[n], {n, 2, 30}]]
%t A208895 f[p_, e_] := p^(3*e) * (1-1/p^2); f[2, e_] := 8^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Oct 18 2022 *)
%o A208895 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 8^f[i,2], f[i,1]^(3*f[i,2]) * (1 - 1/f[i,1]^2))); } \\ _Amiram Eldar_, Oct 18 2022
%Y A208895 Cf. A060968, A087784.
%K A208895 nonn,mult
%O A208895 1,2
%A A208895 _José María Grau Ribas_, Mar 03 2012