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

1,2

Comments

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,...

Links

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

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A114078 A031704 A173275 * A281397 A206122 A259293

Adjacent sequences: A301864 A301865 A301866 * A301868 A301869 A301870

Keyword

nonn

Author

Amiram Eldar, Mar 28 2018

Status

approved