A015710 Least k >= 0 such that phi(n) * sigma(n) + k^2 is a perfect square, or -1 if impossible.

0, 1, 1, -1, 1, 1, 1, 2, -1, 3, 1, 3, 1, 0, 2, 29, 1, -1, 1, 5, 4, 1, 1, 2, 26, 5, 3, 2, 1, 0, 1, 4, 1, 6, 2, 8, 1, 3, 5, 2, 1, 2, 1, 1, 8, 4, 1, 15, -1, 16, 0, 7, 1, 7, 6, 6, 6, 9, 1, 4, 1, 6, 10, 119, 8, 6, 1, 8, 1, 5, 1, 9, 1, 11, 9, 1, 4, 8, 1, 17, -1, 1

Offset: 1

Author

Robert G. Wilson v

Keywords

• sign

Links

• Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Crossrefs

Cf. A015713 (a(n) is zero), A062354 (phi(n)*sigma(n)).

Programs

Mathematica

a[n_] := Module[{m = EulerPhi[n]*DivisorSigma[1, n]}, If[Mod[m, 4] == 2, -1, k = 0; While[!IntegerQ[Sqrt[m + k^2]], k++]; k]]; Array[a, 100] (* Amiram Eldar, Dec 07 2018 *)

PARI

a(n) = {my(x = sigma(n)*eulerphi(n)); if ((x % 4) == 2, -1, my(k=0); while (! issquare(x+k^2), k++); k;);} \\ Michel Marcus, Dec 07 2018

Extensions

a(14), a(30), and a(51) corrected by Sean A. Irvine, Dec 06 2018

Entry revised by Amiram Eldar, Sean A. Irvine, and Michel Marcus, Dec 06 2018