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

Author

Amiram Eldar, Oct 10 2019

Keywords

• nonn
• fini

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.

Examples

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

PARI

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

Extensions

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