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

1,8

Links

Table of n, a(n) for n=1..82.

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A258643 A287561 A046924 * A108415 A278572 A136644

Adjacent sequences: A015707 A015708 A015709 * A015711 A015712 A015713

Keyword

sign

Author

Robert G. Wilson v

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

Status

approved