A328276 -id:A328276 - cp's OEIS frontend

A328274 Numbers m such that phi(m) = rad(m)^3, where phi is the Euler totient function (A000010) and rad is the squarefree kernel function (A007947).

1, 16, 648, 2500, 101250, 259308, 1542294, 3660250, 4691556, 33734898, 846948966, 1877404326, 21107677374, 39691260010, 535377671178, 178726991395974

Offset: 1

Amiram Eldar, Oct 10 2019

De Koninck et al. showed that there are 16 terms in this sequence.

			16 is in the sequence since phi(16) = 8, rad(16) = 2 and 8 = 2^3.

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.

Cf. A000010, A007947, A105261, A211413, A328275, A328276.

rad[n_] := Times @@ First /@ FactorInteger[n]; aQ[n_] := EulerPhi[n] == rad[n]^3; Select[Range[5*10^6], aQ]

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).

Original entry on oeis.org

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

Amiram Eldar, Oct 10 2019

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.

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

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.

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

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

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

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

cp's OEIS Frontend

A328274 Numbers m such that phi(m) = rad(m)^3, where phi is the Euler totient function (A000010) and rad is the squarefree kernel function (A007947).

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