A350059 Numbers k such that 3k and 4k have the same number of divisors.

2, 10, 14, 22, 24, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98, 106, 110, 118, 120, 122, 130, 134, 142, 146, 154, 158, 166, 168, 170, 178, 182, 190, 194, 202, 206, 214, 218, 226, 230, 238, 242, 250, 254, 262, 264, 266, 274, 278, 286, 288, 290, 298, 302, 310, 312, 314

Offset

1,1

Comments

Includes all numbers whose prime factorization has one 2 and no 3's, or three 2's and one 3.

Numbers k such that v_2(k) - 2*v_3(k) = 1, where v_p(k) is the p-adic valuation of k. - Amiram Eldar, Dec 12 2021

Numbers of the form 2 * 12^k * A007310(m) for k >= 0 and m >= 1. - David A. Corneth, Dec 12 2021

The asymptotic density of this sequence is 2/11. - Amiram Eldar, Feb 02 2024

Links

Winston de Greef, Table of n, a(n) for n = 1..10000

Example

30 is not in the sequence: 30*3=90 has 12 divisors, but 30*4=120 has 16 divisors.

Mathematica

Select[Range[300], IntegerExponent[#, 2] - 2 * IntegerExponent[#, 3] == 1 &] (* Amiram Eldar, Dec 12 2021 *)

Prog

(PARI) isok(k) = numdiv(3*k) == numdiv(4*k); \\ Michel Marcus, Dec 12 2021
(PARI) isok(n) = valuation(n, 2) - 2 * valuation(n, 3) == 1; \\ Amiram Eldar, Feb 02 2024
(Python)
from sympy import multiplicity as v
def ok(n): return v(2, n) - 2*v(3, n) == 1
print([k for k in range(1, 315) if ok(k)]) # Michael S. Branicky, Feb 02 2024

Crossrefs

Cf. A000005, A007310, A007814, A007949.

Sequence in context: A050546 A032384 A032624 * A349833 A091999 A230624

Adjacent sequences: A350056 A350057 A350058 * A350060 A350061 A350062

Keyword

nonn,easy

Author

J. Lowell, Dec 11 2021

Extensions

More terms from Michel Marcus, Dec 12 2021

Status

approved