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

%I

%S 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,

%T 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,

%U 119,8,6,1,8,1,5,1,9,1,11,9,1,4,8,1,17,-1,1

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

%H Richard K. Guy, <a href="https://www.jstor.org/stable/2974586">Divisors and desires</a>, Amer. Math. Monthly, 104 (1997), 359-360.

%t 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 *)

%o (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

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

%K sign

%O 1,8

%A _Robert G. Wilson v_

%E a(14), a(30), and a(51) corrected by _Sean A. Irvine_, Dec 06 2018

%E Entry revised by _Amiram Eldar_, _Sean A. Irvine_, and _Michel Marcus_, Dec 06 2018

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Last modified October 1 03:36 EDT 2020. Contains 337441 sequences. (Running on oeis4.)