A369210 Numbers k such that the number of divisors of k^2 is a power of 3.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Author

Amiram Eldar, Jan 16 2024

Keywords

• nonn
• easy

Comments

First differs from A197680 at n = 331, from A274034 at n = 42, from A361177 at n = 167, and from A366762 at n = 84.

Equivalently, square roots of the numbers whose number of divisors is a power of 3.

The asymptotic density of this sequence is Product_{p prime} ((1 - 1/p) * Sum_{k>=0} 1/p^((3^k-1)/2)) = 0.64033435998103973346... .

Links

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

Crossrefs

Cf. A000005, A000244, A000290.

Cf. A197680, A274034, A361177, A366762.

Programs

Mathematica

pow3q[n_] := n == 3^IntegerExponent[n, 3]; Select[Range[100], pow3q[DivisorSigma[0, #^2]] &]

PARI

ispow3(n) = n == 3^valuation(n, 3);
is(n) = ispow3(numdiv(n^2));

Formula

Sum_{n>=1} 1/a(n)^2 = Product_{p prime} Sum_{k>=0} 1/p^(3^k-1) = 1.52478035628964060288... .