A067781 -id:A067781 - cp's OEIS frontend

A301867 Numbers n such that phi(n), psi(n) and sigma(n) are simultaneously perfect squares.

1, 170, 679, 5770, 9154, 9809, 14322, 22413, 37114, 51455, 82615, 100821, 101153, 115430, 139954, 171069, 198462, 222893, 233358, 270485, 349894, 389045, 391090, 514294, 561782, 595122, 608685, 664853, 701338, 815538, 1009527, 1231230, 1290993, 1397091

Offset: 1

Amiram Eldar, Mar 28 2018

nonn

Subsequence of A067781.

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,...

			170 is in the sequence since phi(170) = 64 = 8^2, sigma(170) = psi(170) = 324 = 18^2.
25264008, the first nonsquarefree term ( > 1 ) in the sequence, has phi = 2520^2, sigma = 9600^2 and psi = 8064^2.

Amiram Eldar, Table of n, a(n) for n = 1..1000

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

a = {}; psi[n_] := DirichletConvolve[j, MoebiusMu[j]^2, j, n];
aQ[n_] := AllTrue[Sqrt[{DivisorSigma[1, n], EulerPhi[n], psi[n]}], IntegerQ];
Do[If[aQ[k], AppendTo[a, k]], {k,1,10^7}]; a

cp's OEIS Frontend

A301867 Numbers n such that phi(n), psi(n) and sigma(n) are simultaneously perfect squares.

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