A240547 - OEIS

%I #43 Oct 18 2022 07:27:22

%S 1,8,33,32,145,264,385,128,945,1160,1441,1056,2353,3080,4785,512,5185,

%T 7560,7201,4640,12705,11528,12673,4224,18625,18824,26001,12320,25201,

%U 38280,30721,2048,47553,41480,55825,30240,51985,57608,77649,18560,70561,101640

%N Number of non-congruent solutions of x^2 + y^2 + z^2 + t^2 == 0 mod n.

%H David A. Corneth, <a href="/A240547/b240547.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 László Tóth, <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.

%H <a href="/index/Su#sums_of_squares">Index to sequences related to sums of squares</a>.

%F Multiplicative, with a(2^e) = 2^(2e+1) for e>=1, a(p^e) = p^(2e-1)*(p^(e+1)+p^e-1) for p > 2, e>=1.

%F For odd n, a(n) = A069097(n)*n = A020478(n). - _R. J. Mathar_, Jun 23 2018

%F Sum_{k=1..n} a(k) ~ c * n^4 + O(n^3 * log(n)), where c = 5*Pi^2/(168*zeta(3)) = 0.244362... (Tóth, 2014). - _Amiram Eldar_, Oct 18 2022

%e For n=2 the a(2)=8 solutions are (0,0,0,0), (1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1), (0,0,1,1), (1,1,1,1).

%p A240547 := proc(n) local a, x, y, z, t ; a := 0 ; for x from 0 to n-1 do for y

%p from 0 to n-1 do for z from 0 to n-1 do for t from 0 to n-1 do if

%p (x^2+y^2+z^2+t^2) mod n = 0 mod n then a := a+1 ; fi; od; od ; od; od;

%p a ; end proc;

%p # alternative

%p A240547 := proc(n)

%p     a := 1;

%p     for pe in ifactors(n)[2] do

%p         p := op(1,pe) ;

%p         e := op(2,pe) ;

%p         if p = 2 then

%p             a := a*p^(2*e+1) ;

%p         else

%p             a := a* p^(2*e-1)*(p^(e+1)+p^e-1) ;

%p         end if;

%p     end do:

%p     a ;

%p end proc:

%p seq(A240547(n),n=1..100) ; # _R. J. Mathar_, Jun 25 2018

%t b[2, e_] := 2^(2 e + 1);

%t b[p_, e_] := p^(2 e - 1)*(p^(e + 1) + p^e - 1);

%t a[n_] := Times @@ b @@@ FactorInteger[n];

%t Array[a, 42] (* _Jean-François Alcover_, Dec 05 2017 *)

%o (PARI) a(n) = my(m); if( n<1, 0, forvec( v = vector(4, i, [0, n-1]), m += (0 == norml2(v)%n))); m /* _Michael Somos_, Apr 07 2014 */

%o (PARI) a(n) = {my(f = factor(n), res = 1, start = 1, p, e, i); if(n % 2 == 0, res = 1<<(f[1,2]<<1+1); start = 2); for(i = start, #f~, p = f[i, 1]; e = f[i, 2]; res*=(p^(e<<1-1)*(p^(e+1)+p^e-1))); res} \\ _David A. Corneth_, Jul 22 2018

%Y Cf. A020478, A069097, A086933, A087687, A208895, A229179.

%K nonn,mult

%O 1,2

%A _Laszlo Toth_, Apr 07 2014