cp's OEIS Frontend

Let rad = A007947, p = A119288, q = A053669, g = A006530, and r = A003557.

Numbers k in A126706 such that p <= r < q < g.

Terms are products k of a number s in A033845 and a number t in A007310 with at least one prime power factor p^m | k such that m > 1.

A divisor d | k is said to be coreful if rad(d) = rad(k), where rad = A007947.

In other words, half the number of divisors d | k such that both gcd(d, k/d) > 1 and rad(d) != rad(k/d).

Divisors d and k/d have at least 1 prime factor in common and at least one prime factor q divides one but not the other divisor. Thus, the reference domain S is the intersection of nonsquarefree numbers (k in A013929) and numbers that are not prime powers (k in A024619).

Let S = { prime p : p | d } and let T = { prime p : p | k/d }. Then this sequence counts divisor pairs (d, k/d), d < k/d, such that the symmetric difference of S and T is not empty. For instance, for k = 24 = 2*12 = 4*6, where, in both cases, the product P of the symmetric difference is 3. For k = 180 = 2*90 = 3*60 = 6*30 = 10*18 = 12*15, the products of symmetric differences are 15, 10, 5, 15, and 10, respectively. In the case of 10*18, it is evident that neither rad(10) = rad(180) nor rad(18) = rad(30).

Primes p < 11 are in the sequence since the smallest number in A126706 is 12.

Consider p > 11, odd primes; then both p-1 and p+1 are even. Let j and k be neighbors of p. One neighbor, j, is also divisible by 4, while the other neighbor k is not divisible by 2^m, m > 1. The latter statement implies k cannot be a perfect power q^m, p != q, m > 0, but q^m may divide k.

This sequence is that of primes where j = 2^m and k is squarefree.

Proper subset of A141453.

The neighbor k is also divisible by 3, since abs(p-k) = 1 and neither are divisible by 3. Therefore, 6 | k.

This is the union of 1 and A024619. It is the sequence C used in the definition of A360519. Since C is central to the analysis of A360519 it deserves its own entry.

This has the same relationship to A024619 as A000469 does to A120944 for squarefree numbers.

Although 1 is a square, is a cube, and so on..., 1 is sometimes excluded from perfect powers since it is not a well-defined power of 1 (1 = 1^k for any k in [2, 3, 4, 5, ...])

From Michael De Vlieger, Aug 11 2025: (Start)

This sequence is A001597 \ A246547, i.e., perfect powers without proper prime powers.

Union of {1} with the intersection of A001597 and A126706, where A126706 is the sequence of numbers that are neither prime powers nor squarefree.

Union of {1} and A286708 \ A052486, i.e., powerful numbers that are not prime powers, without Achilles numbers, but including the empty product. (End)

In other words, C contains all positive numbers except powers of primes p^k, k>=1.

This is a modified version of the Enots Wolley sequence A336957. The modification ensures that the sequence does not contain the prime 2.

Let Ker(k), the kernel of k, denote the set of primes dividing k. Thus Ker(36) = {2,3}, Ker(1) = {}.

Theorem: a(1)=1, a(2)=6; thereafter, a(n) is the smallest number m not yet in the sequence such that

(i) Ker(m) intersect Ker(a(n-1)) is nonempty,

(ii) Ker(m) intersect Ker(a(n-2)) is empty, and

(iii) The set Ker(m) \ Ker(a(n-1)) is nonempty.

Conjecture: The sequence is a permutation of C.

We define a factorization of n into factors > 1 to be co-balanced if it has exactly A001221(n) factors.

Similar to the EKG sequence A064413, but whereas in that sequence a(n) is chosen to be as small as possible, here the primary goal is to choose a(n) to be less than a(n-1) and as close to it as possible. This sequence first differs from the EKG sequence at n = 8, where a(8) = k = 10 is closer to a(7) = 12 than A064413(8) = 8 is.

A significant difference from the EKG sequence is that the primes do not appear in their natural order. Also, it is not always true that a prime p is preceded by 2*p when it first appears. 4k+3 primes appear to be preceded by smaller multiples than 4k+1 primes.

It is conjectured that every positive number appears.

It is interesting to study what happens if the first two terms are taken to be 1,s, with s >= 2, or if the first s terms are taken to be 1,2,3,...,s, with s >= 2. Call two such sequences equivalent if they eventually merge. The 1,3 and 1,2,3 sequences merge with each other after half-a-dozen terms. But at present we do not know if they merge with the 1,2 sequence.

It appears that many sequences that start 1,s and 1,2,3,...,s with small s merge with one of the sequences 1,2 or 1,2,3 or 1,2,3,...,11.

[The preceding comments are from Geoffrey Caveney's emails.]

From Michael De Vlieger, Aug 15 2025: (Start)

There are long runs of terms with the same parity in this sequence. For example, beginning at a(481) = 948, there are 100 consecutive even terms. Starting with a(730076) = 1026330, there are 100869 consecutive even terms, followed by 36709 consecutive odd terms. Runs of even terms tend to be longer than those of odd.

There are long runs of first differences of -2 and -6 in this sequence, and that there appear to be three phases. The predominant (A) phase has a(n) = a(n-1)-2, the second (B) phase has a(n) = a(n-1)-6, and then there is a turbulent (C) phase [C] with varied differences.

Generally the even runs correspond to differences a(n)-a(n-1) = 2 and feature square-free terms separated by an odd number of terms in A126706. Phase [C] tends to be largely odd squarefree semiprimes and includes prime powers. (End)

Sometimes nonsquarefree numbers are misnamed squareful numbers (see 1st comment of A013929). Indeed, every squareful number > 1 is nonsquarefree, but the converse is false. This sequence = A013929 \ A001694 and consists of these counterexamples.

This sequence is not a duplicate: the first 16 terms (<= 68) are the same first 16 terms of A059404, A323055, A242416 and A303946, then 72 is the 17th term of these 4 sequences. Also, the first 37 terms (<= 140) are the same first 37 terms of A317616 then 144 is the 38th term of this last sequence.

From Amiram Eldar, Sep 17 2023: (Start)

Called "hybrid numbers" by Jakimczuk (2019).

These numbers have a unique representation as a product of two numbers > 1, one is squarefree (A005117) and the other is powerful (A001694).

Equivalently, numbers k such that A055231(k) > 1 and A057521(k) > 1.

Equivalently, numbers that have in their prime factorization at least one exponent that is equal to 1 and at least one exponent that is larger than 1.

The asymptotic density of this sequence is 1 - 1/zeta(2) (A229099). (End)

We define a factorization of n into factors > 1 to be cross-balanced if either (1) it is empty or (2) the maximum image of A001222 over the factors is A001221(n).