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A087782 a(n) = number of solutions to x^3 + x == 0 (mod n).

%I A087782 #18 Sep 20 2019 08:55:28
%S A087782 1,2,1,1,3,2,1,1,1,6,1,1,3,2,3,1,3,2,1,3,1,2,1,1,3,6,1,1,3,6,1,1,1,6,
%T A087782 3,1,3,2,3,3,3,2,1,1,3,2,1,1,1,6,3,3,3,2,3,1,1,6,1,3,3,2,1,1,9,2,1,3,
%U A087782 1,6,1,1,3,6,3,1,1,6,1,3,1,6,1,1,9,2,3,1,3,6,3,1,1,2,3,1,3,2,1,3,3,6,1,3,3
%N A087782 a(n) = number of solutions to x^3 + x == 0 (mod n).
%C A087782 Shadow transform of A034262. - _Michel Marcus_, Jun 06 2013
%H A087782 Andrew Howroyd, <a href="/A087782/b087782.txt">Table of n, a(n) for n = 1..10000</a>
%H A087782 Lorenz Halbeisen and Norbert Hungerbuehler, <a href="https://www.semanticscholar.org/paper/Number-theoretic-aspects-of-a-combinatorial-Halbeisen-Hungerb%C3%BChler/5743ff2f9c14d22d1a9e570d6951a7c9ef79a612">Number theoretic aspects of a combinatorial function</a>, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.
%H A087782 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%F A087782 Multiplicative with a(2^1) = 2, a(2^e) = 1 for e > 1, a(p^e) = 3 for p mod 4 == 1, a(p^e) = 1 for p mod 4 == 3. - _Andrew Howroyd_, Jul 15 2018
%t A087782 a[n_] := If[n == 1, 1, Product[{p, e} = pe; If[p == 2, If[e == 1, 2, 1], If[Mod[p, 4] == 1, 3, 1]], {pe, FactorInteger[n]}]];
%t A087782 a /@ Range[1, 100] (* _Jean-François Alcover_, Sep 20 2019 *)
%o A087782 (PARI) a(n)={my(v=vector(n)); sum(i=0, n-1, lift(Mod(i,n)^3 + i) == 0)} \\ _Andrew Howroyd_, Jul 15 2018
%o A087782 (PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); if(p==2, if(e==1, 2, 1), if(p%4==1, 3, 1)))} \\ _Andrew Howroyd_, Jul 15 2018
%Y A087782 Cf. A087688, A000089, A034262.
%K A087782 mult,nonn
%O A087782 1,2
%A A087782 Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003
%E A087782 More terms from _David Wasserman_, Jun 17 2005