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

Author

Amiram Eldar, May 16 2023

Keywords

• nonn

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).

Examples

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

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.

Crossrefs

Cf. A000005, A000010, A048691, A062821.

Cf. A052291, A259021.

Programs

Mathematica

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

PARI

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