cp's OEIS Frontend

Numbers k whose largest unitary divisor that is a square, A350388(k), is a prime power (A246655), or equivalently, A350388(k) is in A056798 \ {1}.

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

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

The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} p/((p-1)*(p+1)^2) = 0.24200684327095676029... .

Numbers whose sorted prime signature (A118914) is either of the form {2*p-1} or {1, p-1}, where p is a prime. Equivalently, disjoint union of numbers of the form q^(2*p-1) where p and q are primes, and numbers of the form r * q^(p-1), where p, q and r are primes and r != q. - Amiram Eldar, Sep 09 2024

All terms are of the form p^4 * q, with primes p < q.

a(3) = 13203 = 3^4 * 163 is the only term for which q = 2*p^4 + 1; for all other terms, q is either 2*p^4 - 1 (e.g., a(1) = 496 = 2^4 * 31) or (p^4 + 1)/2 (e.g., a(2) = 3321 = 3^4 * 41).

If d(j(n)) is prime p then d(a(n+1)) must be properly divisible by p. In practice the proper divisor for computation of a(n+1) toggles between d(j(n)) and d(k).

Conjecture: This is a permutation of the positive integers. Numbers with the same number (tau) of divisors appear in their natural orders (e.g., primes, semiprimes, squares).

The plot, after the first few terms, resolves itself into points tightly packed on and around a straight line of slope 1, with exceptional points appearing as significant upward or downward "spikes".

When d(j(n)) is prime p appearing for the first time in the sequence J = {d(j(a(n)), n>=1}, then a(n+1) is the smallest number with 2p divisors, which produces a significantly large upward spike above the straight line (6, 12, 48, 192, 3072, 12288, ...).

When d(j(a(n)) is 2p, seen for the first time in J, then a(n+1) is the smallest number with p divisors, which produces a large downward spike, below the straight line (2, 4, 16, 64, 1024, 4096, ...).

The sequence of fixed points starts: 1, 46, 69, 74, 110, 140, 142, 152, 154, 178, ... apparently becoming denser as n increases.

First differs from A344653 and A345193 at n = 17: a(17) = 120 is not a term of these sequences.

Numbers whose prime signature (A118914) is of the form {1, 1, ..., m} with m >= 3, i.e., any number (including zero) of 1's and then a single number >= 3.

The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/(p*(p^2-1)) = A369632 / A013661 = 0.13463358553764438661... .

There is a strong correlation between values of this function and values of other arithmetic functions. In other words, a(n) correlates to a single distinct value from one or more of the arithmetic functions.

Terms of this sequence select from the positive integers as follows:

A318366(k) = a(1), 1 followed by the primes (A008578).

A318366(k) = A008836(k) = A001221(k) = a(2), primes squared (A001248).

A318366(k) = A001221(k) = a(3), squarefree semiprimes (A006881).

A318366(k) = A000005(k) = a(4), primes cubed (A030078).

A318366(k) = a(5), a prime squared times a prime (A054753).

A318366(k) = a(6), primes to the fourth power (A030514).

A318366(k) = a(7), sphenic numbers (A007304).

A318366(k) = a(8), union of A050997 and A065036.

A318366(k) = a(9), squarefree semiprimes squared (A085986).

A318366(k) = a(10), product of four primes, three distinct (A085987).

A318366(k) = a(11), primes to the sixth power (A030516).

A318366(k) = a(12), product of prime to fourth power and a different prime (A178739).

A318366(k) = a(13), product of four distinct primes (A046386).

...

Intersection of A319240 and A013929, i.e., terms of A319240 that are not squarefree.

A319240 lists the numbers that have an equal number of factorizations of even and odd length in an unordered manner.

A013929 lists the numbers that have an equal number of factorizations of even and odd length in an ordered manner.

There are infinitely many terms in this sequence since p^2 is always such a number for prime p.

Out of all numbers of the form p^k with p prime (listed in A000961), only the numbers of the form p^2 (A001248) are terms.

Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881), p*q^2 (A054753), p*q^4 (A178739) and p*q^6 (A189987) are terms.

Similar methods can be applied to all prime signatures.

Wilf's conjecture is equivalent to the statement that this sequence is the set difference of A319240 and A006881.