A328275 Numbers m such that phi(m) = rad(m)^4, where phi is the Euler totient function (A000010) and rad is the squarefree kernel function (A007947).

1, 32, 3888, 25000, 2839714, 3037500, 10890936, 120298932, 402627500, 534837384, 7489147356, 8508543750, 48919241250, 111945866022, 336977358354, 417841706250, 553904623764, 1498168652148, 2627525125250, 2761526809032, 2898701538750, 7978057537338, 16548448068126, 20978349935382

Offset

1,2

Comments

De Koninck et al. showed that there are 85 terms in this sequence, yet a(6) = 3037500 was missing in their paper. With a(6), it was verified numerically that the first 38 terms (terms below 10^18) are correct.

There are no other missing terms and the sequence contains 86 terms overall. - Max Alekseyev, Sep 30 2025

Links

Max Alekseyev, Table of n, a(n) for n = 1..86

Jean-Marie De Koninck, Florian Luca and A. Sankaranarayanan, Positive integers whose Euler function is a power of their kernel function, Rocky Mountain Journal of Mathematics, Vol. 36, No. 1 (2006), pp. 81-96, alternative link.

Example

32 is in the sequence since phi(32) = 16, rad(32) = 2 and 16 = 2^4.

Mathematica

rad[n_] := Times @@ First /@ FactorInteger[n]; aQ[n_] := EulerPhi[n] == rad[n]^4; Select[Range[3*10^6], aQ]

Prog

(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
isok(m) = eulerphi(m) == rad(m)^4; \\ Michel Marcus, Oct 15 2019

Crossrefs

Cf. A000010, A007947, A105261, A211413, A328274, A328276.

Sequence in context: A264126 A304401 A394620 * A291829 A248075 A248000

Adjacent sequences: A328272 A328273 A328274 * A328276 A328277 A328278

Keyword

nonn,fini,full

Author

Amiram Eldar, Oct 10 2019

Extensions

a(6) = 3037500 from Marius A. Burtea, Oct 11 2019

Status

approved