cp's OEIS Frontend

If k = Product p^e, then A162296(k) / A048250(k) = -1 + Product (p^(e+1) - 1)/(p^2 - 1). If k is exponentially odd, then e = 2*m - 1 is odd for all the prime factors p of k and p^(e+1) - 1 = (p^2)^m - 1 is divisible by p^2 - 1. Therefore, A162296(k) / A048250(k) is an integer for all exponentially odd numbers, and it is a positive integer for all the nonsquarefree (A013929) exponentially odd numbers.

It seems that most of the terms of A301517 are exponentially odd numbers. For example, the first 10^4 terms of A301517 include only 9 terms that are not exponentially odd numbers. Up to 10^8 there are 9660732 terms of A301517, and only 9107 of them are not exponentially odd numbers.

The number of terms of this sequence that do not exceed 10^k, for k = 5, 6, ... are 9, 92, 916, 9107, 91172, 911187, .... Apparently, this sequence has an asymptotic density c = 0.000091... If this is true, then the asymptotic density of A301517 is c + A065463 - A059956 = 0.096606... (A065463 is the density of the exponentially odd numbers, and A059956 is the density of the squarefree numbers which are a subset of the exponentially odd numbers).

First differs from A301517 at n = 1213. A301517(1213) = 12500 = 2^2 * 5^5 is not an exponentially odd number.

Numbers whose exponents in their prime factorization are all odd and at least one of them is larger than 1.

All the squarefree numbers (A005117) are exponentially odd. Therefore, the sequence of exponentially odd numbers (A268335) is a disjoint union of the squarefree numbers and this sequence.

The asymptotic density of this sequence is A065463 - A059956 = 0.096515099145... .

Subsequence of A301517 and A374459 and first differs from them at n = 21. A301517(21) = A374459(21) = 216 is not a term of this sequence.

Numbers having exactly one non-unitary prime factor and its multiplicity is odd.

Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m+1} with m >= 1, i.e., any number (including zero) of 1's and then a single odd number > 1.

The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/((p-1)*(p+1)^2) = 0.093382464285953613312...

The asymptotic density of this sequence is A065463 - A059956 = 0.09651509914... . - Amiram Eldar, Sep 27 2023

Subsequence of A301517 and A374459 and first differs from them at n = 85: A374459(85) = A374459(85) = 864 = 2^5 * 3^3 is not a term of this sequence.

First differs from its subsequence A381312 at n = 21: a(21) = 216 = 2^3 * 3^3 is not a term of A381312.

Numbers whose prime factorization has one distinct exponent that is larger than 1 and it is odd.

Numbers that are a product of a squarefree number (A005117) and a coprime nonsquarefree number that is a squarefree number raised to an odd power (A384518).

The asymptotic density of this sequence is Sum_{k>=1} (d(2*k+1)-1)/zeta(2) = 0.095609588748823080455..., where d(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i).