A363059 Numbers k such that the number of divisors of k^2 equals the number of divisors of phi(k), where phi is the Euler totient function.

1, 5, 57, 74, 202, 292, 394, 514, 652, 1354, 2114, 2125, 3145, 3208, 3395, 3723, 3783, 4053, 4401, 5018, 5225, 5298, 5425, 5770, 6039, 6363, 6795, 6918, 7564, 7667, 7676, 7852, 7964, 8585, 9050, 9154, 10178, 10535, 10802, 10818, 10954, 11223, 12411, 13074, 13634

Offset

1,2

Comments

Numbers k such that A048691(k) = A062821(k).

Amroune et al. (2023) characterize solutions to this equation and prove that Dickson's conjecture implies that this sequence is infinite.

They show that the only squarefree semiprime terms are 57, 514 and some of the numbers of the form 2*(4*p^2+1), where p and 4*p^2+1 are both primes (a subsequence of A259021).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Zahra Amroune, Djamel Bellaouar and Abdelmadjid Boudaoud, A class of solutions of the equation d(n^2) = d(phi(n)), Notes on Number Theory and Discrete Mathematics, Vol. 29, No. 2 (2023), pp. 284-309.

Wikipedia, Dickson's conjecture.

Example

5 is a term since both 5^2 = 25 and phi(5) = 4 have 3 divisors.

Mathematica

Select[Range[15000], DivisorSigma[0, #^2] == DivisorSigma[0, EulerPhi[#]] &]

Prog

(PARI) is(n) = numdiv(n^2) == numdiv(eulerphi(n));

Crossrefs

Cf. A000005, A000010, A048691, A062821.

Cf. A052291, A259021.

Sequence in context: A180874 A336243 A356134 * A163793 A226425 A076455

Adjacent sequences: A363056 A363057 A363058 * A363060 A363061 A363062

Keyword

nonn

Author

Amiram Eldar, May 16 2023

Status

approved