A328274 -id:A328274 - cp's OEIS frontend

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

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

A328276 The number of solutions to phi(x) = rad(x)^n, where phi is the Euler totient function (A000010) and rad is the squarefree kernel function (A007947).

Original entry on oeis.org

3, 6, 16, 85, 969

Offset: 1

Amiram Eldar, Oct 10 2019

The author informs me that because there of an error in one of the references, a(4) is > 85. - N. J. A. Sloane, Nov 23 2019

			a(1) = 3 since there are only 3 solutions to phi(x) = rad(x): x = 1, 4, and 18.
a(2) = 6 since there are only 6 solutions to phi(x) = rad(x)^2: x = 1, 8, 108, 250, 6174, and 41154 (the terms of A211413).

Jean-Marie De Konick and Stefan Gubo, When the totient is the product of the squared prime divisors: problem 10966, American Mathematical Monthly, Vol. 111, No. 6 (2004), p. 536.
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, A328275.

cp's OEIS Frontend

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).

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