A382061 Numbers whose number of divisors is divisible by their number of unitary divisors.

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97

Offset: 1

Author

Amiram Eldar, Mar 14 2025

Keywords

• nonn
• easy

Comments

Numbers k such that A034444(k) | A000005(k).

The criterion according to which a number belongs to this sequence depends only on the prime signature of this number: if {e_1, e_2, ... } are the exponents in the prime factorization of k then k is a term if and only if A000005(k)/A344444(k) = Product_{i} (e_i + 1)/2 is an integer.

The exponentially odd numbers (A268335) are all terms, since their prime factorization has only odd exponents e_i, so (e_i + 1)/2 is an integer. This sequence first differs from A268335 at n = 53: a(53) = 72 = 2^3 * 3^2 is not a term of A268335. The next terms that are not in A268335 are 108, 200, 360, 392, 432, 500, ... .

All the squarefree numbers (A005117, which is a subsequence of A268335) are terms. These are the numbers k such that A034444(k) = A000005(k).

A number k is a term if and only if the powerful part of k, A057521(k), is a term. Therefore, the primitive terms of this sequence are the powerful terms, A382062.

The asymptotic density of this sequence is Sum_{n>=1} f(A382062(n)) = 0.72201619..., where f(n) = (n/zeta(2)) * Product_{prime p|n} (p/(p+1)).

The asymptotic density of a few subsequences can be evaluated more easily. For example:

1) Powerful numbers that are exponentially odd (A335988): When summing only over these numbers, the formula for the asymptotic density gives the density of the exponentially odd numbers: Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463).

2) Numbers of the form p^(2*k) * q^(2*m+1), where k and m >= 1, and p != q are primes: When summing only over these numbers, the density of the numbers whose powerful part is of this form is ((Sum_{p prime} p/((p^2-1)*(p+1))) * (Sum_{p prime} p^2/((p^4-1)*(p+1))) - Sum_{p prime} p^3/((p^2-1)^2*(p^2+1)*(p+1)^2)) / zeta(2) = 0.017174455422470834821... .

Links

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

Crossrefs

Cf. A000005, A013661, A034444, A057521, A065463, A382063.

Subsequences: A005117, A268335, A382062.

Programs

Mathematica

q[k_] := Divisible[DivisorSigma[0, k], 2^PrimeNu[k]]; Select[Range[100], q]

PARI

isok(k) = {my(f = factor(k)); !(numdiv(f) % (1<
				

Formula

2 is a term since A000005(2) = A034444(2) = 2, so 2 | 2.

24 is a term since A000005(24) = 8, A034444(24) = 4, and 4 | 8.