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Numbers n such that phi(n), psi(n) and sigma(n) are simultaneously perfect squares.
1

%I #12 Aug 19 2019 03:16:54

%S 1,170,679,5770,9154,9809,14322,22413,37114,51455,82615,100821,101153,

%T 115430,139954,171069,198462,222893,233358,270485,349894,389045,

%U 391090,514294,561782,595122,608685,664853,701338,815538,1009527,1231230,1290993,1397091

%N Numbers n such that phi(n), psi(n) and sigma(n) are simultaneously perfect squares.

%C Subsequence of A067781.

%C The squarefree terms of A067781 are in this sequence (since if n is squarefree then psi(n) = sigma(n)). The nonsquarefree terms of this sequence are: 1, 25264008, 2171889720, 2659240584, 5893922664, 16962447600,...

%H Amiram Eldar, <a href="/A301867/b301867.txt">Table of n, a(n) for n = 1..1000</a>

%e 170 is in the sequence since phi(170) = 64 = 8^2, sigma(170) = psi(170) = 324 = 18^2.

%e 25264008, the first nonsquarefree term ( > 1 ) in the sequence, has phi = 2520^2, sigma = 9600^2 and psi = 8064^2.

%t a = {}; psi[n_] := DirichletConvolve[j, MoebiusMu[j]^2, j, n];

%t aQ[n_] := AllTrue[Sqrt[{DivisorSigma[1, n], EulerPhi[n], psi[n]}], IntegerQ];

%t Do[If[aQ[k], AppendTo[a, k]], {k,1,10^7}]; a

%Y Cf. A000010 (phi), A000203 (sigma), A001615 (psi), A067781, A291549.

%K nonn

%O 1,2

%A _Amiram Eldar_, Mar 28 2018