The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A087687 Number of solutions to x^2 + y^2 + z^2 == 0 (mod n). 5
 1, 4, 9, 8, 25, 36, 49, 32, 99, 100, 121, 72, 169, 196, 225, 64, 289, 396, 361, 200, 441, 484, 529, 288, 725, 676, 891, 392, 841, 900, 961, 256, 1089, 1156, 1225, 792, 1369, 1444, 1521, 800, 1681, 1764, 1849, 968, 2475, 2116, 2209, 576, 2695, 2900, 2601 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS To show that a(n) is multiplicative is simple number theory. If gcd(n,m)=1, then any solution of x^2 + y^2 + z^2 == 0 (mod n) and any solution (mod m) can combined to a solution (mod nm) using the Chinese Remainder Theorem and any solution (mod nm) gives solutions (mod n) and (mod m). Hence a(nm) = a(n)*a(m). - Torleiv Kløve, Jan 26 2009 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 80 terms from Robert Gerbicz) C. Calderón and M. J. De Velasco, On divisors of a quadratic form, Boletim da Sociedade Brasileira de Matemática, Vol. 31, No. 1 (2000), pp. 81-91; alternative link. László Toth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq., Vol. 17 (2014), Article # 14.11.6; arXiv preprint, arXiv:1404.4214 [math.NT], 2014. FORMULA a(2^k) = 2^(k + ceiling(k/2)). For odd primes p, a(p^(2k-1)) = p^(3k-2)*(p^k + p^(k-1) - 1) and a(p^(2k)) = p^(3k-1)*(p^(k+1) + p^k - 1). - Martin Fuller, Jan 26 2009 Sum_{k=1..n} a(k) ~ (4*zeta(3))/(15*zeta(4)) * n^3 + O(n^2 * log(n)) (Calderón and de Velasco, 2000). - Amiram Eldar, Mar 04 2021 MAPLE A087687 := proc(n)     a := 1;     for pe in ifactors(n)[2] do         p := op(1, pe) ;         e := op(2, pe) ;         if p = 2 then             a := a*p^(e+ceil(e/2)) ;         elif type(e, 'odd') then             a := a*p^((3*e-1)/2)*(p^((e+1)/2)+p^((e-1)/2)-1) ;         else             a := a*p^(3*e/2-1)*(p^(e/2+1)+p^(e/2)-1) ;         end if;     end do:     a ; end proc: seq(A087687(n), n=1..100) ; # R. J. Mathar, Jun 25 2018 MATHEMATICA a[n_] := Module[{k=1}, Do[{p, e} = pe; k = k*If[p == 2, p^(e + Ceiling[ e/2]), If[OddQ[e], p^((3*e-1)/2)*(p^((e+1)/2) + p^((e-1)/2) - 1), p^(3*e/2 - 1)*(p^(e/2 + 1) + p^(e/2) - 1)]], {pe, FactorInteger[n]}]; k]; Array[a, 100] (* Jean-François Alcover, Jul 10 2018, after R. J. Mathar *) PROG (PARI) a(n)=local(v=vector(n), w); for(i=1, n, v[i^2%n+1]++); w=vector(n, i, sum(j=1, n, v[j]*v[(i-j)%n+1])); sum(j=1, n, w[j]*v[(1-j)%n+1]) \\ Martin Fuller (PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); if(p==2, 2^(e+(e+1)\2), p^(e+(e-1)\2)*(p^(e\2)*(p+1) - 1)))} \\ Andrew Howroyd, Aug 06 2018 CROSSREFS Cf. A086933, A062775. Different from A064549. Sequence in context: A073395 A064549 A304203 * A264090 A297439 A168175 Adjacent sequences:  A087684 A087685 A087686 * A087688 A087689 A087690 KEYWORD mult,look,nonn AUTHOR Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 27 2003 EXTENSIONS More terms from Robert Gerbicz, Aug 22 2006 Edited by Steven Finch, Feb 06 2009, Feb 12 2009 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 16 17:22 EDT 2021. Contains 343949 sequences. (Running on oeis4.)