cp's OEIS Frontend

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

%I A382061 #9 Mar 14 2025 21:16:44
%S A382061 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,
%T A382061 34,35,37,38,39,40,41,42,43,46,47,51,53,54,55,56,57,58,59,61,62,65,66,
%U A382061 67,69,70,71,72,73,74,77,78,79,82,83,85,86,87,88,89,91,93,94,95,96,97
%N A382061 Numbers whose number of divisors is divisible by their number of unitary divisors.
%C A382061 Numbers k such that A034444(k) | A000005(k).
%C A382061 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.
%C A382061 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, ... .
%C A382061 All the squarefree numbers (A005117, which is a subsequence of A268335) are terms. These are the numbers k such that A034444(k) = A000005(k).
%C A382061 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.
%C A382061 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)).
%C A382061 The asymptotic density of a few subsequences can be evaluated more easily. For example:
%C A382061 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).
%C A382061 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... .
%H A382061 Amiram Eldar, <a href="/A382061/b382061.txt">Table of n, a(n) for n = 1..10000</a>
%F A382061 2 is a term since A000005(2) = A034444(2) = 2, so 2 | 2.
%F A382061 24 is a term since A000005(24) = 8, A034444(24) = 4, and 4 | 8.
%t A382061 q[k_] := Divisible[DivisorSigma[0, k], 2^PrimeNu[k]]; Select[Range[100], q]
%o A382061 (PARI) isok(k) = {my(f = factor(k)); !(numdiv(f) % (1<<omega(f)));}
%Y A382061 Cf. A000005, A013661, A034444, A057521, A065463, A382063.
%Y A382061 Subsequences: A005117, A268335, A382062.
%K A382061 nonn,easy
%O A382061 1,2
%A A382061 _Amiram Eldar_, Mar 14 2025