A239614 a(n) = A239611(n) / A079458(n).

1, 2, 2, 4, 2, 4, 2, 6, 3, 4, 2, 8, 2, 4, 4, 8, 2, 6, 2, 8, 4, 4, 2, 12, 3, 4, 4, 8, 2, 8, 2, 10, 4, 4, 4, 12, 2, 4, 4, 12, 2, 8, 2, 8, 6, 4, 2, 16, 3, 6, 4, 8, 2, 8, 4, 12, 4, 4, 2, 16, 2, 4, 6, 12, 4, 8, 2, 8, 4, 8, 2, 18, 2, 4, 6, 8, 4, 8, 2, 16, 5, 4, 2, 16, 4, 4, 4, 12, 2, 12, 4, 8, 4, 4, 4, 20, 2, 6, 6, 12, 2, 8, 2, 12, 8

Offset: 1

Author

José María Grau Ribas, Jun 25 2014

Keywords

• nonn
• mult

Comments

Related to Menon's identity. See Conclusions and further work section of the arXiv file linked.

Multiplicative because both A239611 and A079458 are. - Andrew Howroyd, Aug 07 2018

Links

• Antti Karttunen, Table of n, a(n) for n = 1..2101 (computed from the b-files of A079458 and A239611)
• C. Calderón, J. M. Grau, A. Oller-Marcen, and Laszlo Toth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014.

Crossrefs

Cf. A239611, A239612, A239613, A239615, A079458, A053191, A227499.

Programs

Mathematica

a239611[n_] := Sum[If[GCD[x^2 + y^2, n] == 1, GCD[x^2 + y^2 - 1, n], 0], {x, 1, n}, {y, 1, n}];
a079458[n_] := Product[{p, e} = pe; Which[p==2, 2^(2e-1), Mod[p, 4]==3, (p^2-1)p^(2e-2), Mod[p, 4]==1, (p-1)^2 p^(2e-2)], {pe, FactorInteger[n]}];
a[1] = 1; a[n_] := a239611[n]/a079458[n];
Array[a, 105] (* Jean-François Alcover, Dec 04 2018 *)

Formula

Conjectures from Ridouane Oudra, Jul 22 2024: (Start)

a(n) = A010710(n)*tau(n) - 2*tau(2n) ;

a(2*n) = 2*tau(n) ;

a(2*n+1) = tau(2*n+1). (End)

Extensions

More terms from Antti Karttunen, Sep 23 2017