Structural boundaries, partitions and resonance of radical-prime multiplicative sequences

1. Introduction

In multiplicative number theory, the relationship between an integer $k$, its squarefree kernel (the radical), and its divisor-counting functions draws some surprisingly sharp boundaries within the natural numbers. This paper works through both empirical computation and formal proof for sequences where an integer equals the product of its radical and a prime-factor-counting metric.

First, some notation. For any positive integer $k > 1$ with canonical prime factorization $k = p_1^{a_1} p_2^{a_2} \cdots p_m^{a_m}$:

• Little Omega Function ($\omega(k)$ — A001221): the number of distinct prime factors of $k$: $\omega(k) = m = \sum_{p \mid k} 1$
• Big Omega Function ($\Omega(k)$ — A001222): the total number of prime factors, counted with multiplicity: $\Omega(k) = \sum_{i=1}^{m} a_i$
• The Radical ($\text{rad}(k)$ — A007947): the greatest squarefree divisor of $k$, i.e. the product of its distinct prime factors: $\text{rad}(k) = \prod_{i=1}^{m} p_i$
• The Kernel Ratio (A003557): what’s left of $k$ once you strip out the radical — its non-squarefree part: $\frac{k}{\text{rad}(k)} = \prod_{i=1}^{m} p_i^{a_i - 1}$
• Greatest Prime Factor ($\text{gpf}(k)$ — A006530): the largest prime $p_m$ dividing $k$.
• Least Prime Factor ($\text{lpf}(k)$ — A020639): the smallest prime $p_1$ dividing $k$.

With these in hand, we can sort integers into a few useful domains:

1. Squarefree Numbers (A005117): integers with no repeated prime factor. Here $k = \text{rad}(k)$, since every $a_i = 1$, and so $\omega(k) = \Omega(k)$.
2. Powerful Numbers / Squareful Numbers (A001694): every prime factor appears with exponent $a_i \ge 2$. Equivalently, $\text{rad}(k)^2 \mid k$.
3. Achilles Numbers (A052486): powerful numbers that aren’t perfect powers (A001597). These need at least two distinct primes with setwise-coprime exponents greater than 1, giving $\Omega(k) > 2\omega(k)$.
4. Squaremid Numbers (A332785): neither squarefree nor powerful — the middle ground, where at least one prime has multiplicity $\ge 2$ and at least one has multiplicity $1$. These satisfy $\omega(k) < \Omega(k)$ and $\text{rad}(k)^2 \nmid k$.

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2. Structural analysis of sequence A396594

Sequence A396594 is defined by the resonance identity:

$k = \text{rad}(k) \times \Omega(k)$

Call its set of terms $\mathcal{S}$. Scanning the early terms turns up the primes (A000040), the prime square $4 = 2^2$, and a surprisingly regular family of composites.

2.1 Two primary infinite composite families

Set aside $4$ and the primes: every composite term up to $k \le 193$ falls into one of two infinite families, both built by giving one prime factor a small exponent while leaving the other linear.

The $9p$ family ($m = 3^2 \times p$)

Take any prime $p \ne 3$ and let $m = 3^2 \times p$.

• Distinct prime factors are $3$ and $p$, so: $\text{rad}(m) = 3 \times p = 3p$
• Multiplicities are $a_3 = 2$, $a_p = 1$, so: $\Omega(m) = 2 + 1 = 3$
• The resonance product: $\text{rad}(m) \times \Omega(m) = (3p) \times 3 = 9p = m$ So every number of the form $9p$, $p \ne 3$ prime, sits in A396594 — this accounts for terms like $18, 45, 63, 99, 117, 153, \dots$

The $8p$ family ($m = 2^3 \times p$)

Take any odd prime $p$ and let $m = 2^3 \times p$.

• Distinct prime factors are $2$ and $p$: $\text{rad}(m) = 2 \times p = 2p$
• Multiplicities are $a_2 = 3$, $a_p = 1$: $\Omega(m) = 3 + 1 = 4$
• The resonance product: $\text{rad}(m) \times \Omega(m) = (2p) \times 4 = 8p = m$ So every $8p$ with $p$ an odd prime also belongs, giving terms like $24, 40, 56, 88, 104, 136, 152, \dots$

Both families intersect A096156 (numbers of the form $p^2 q$) and A366825 — which is really just telling us why composite terms need at least one exponent equal to 1.

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2.2 Non-conforming composites and high-$\Omega$ trajectories

Call $S_{20260704}$ (cataloged as A396157) the subsequence of composites in A396594 that don’t fit prime powers or the $8p$/$9p$ pattern. It starts:

$1050, 1260, 1650, 1950, 1980, 2340, 2550, 2772, 2850, 3060, 3276, 3420, 3450, 3850, \dots$

Index ($n$)	Term $a(n)$	Factorization	Radical $\text{rad}(a(n))$	$\Omega(a(n))$	$\text{rad} \times \Omega$
1	4	$2^2$	2	2	$2 \times 2 = 4$
238	1050	$2 \times 3 \times 5^2 \times 7$	210	5	$210 \times 5 = 1050$
277	1260	$2^2 \times 3^2 \times 5 \times 7$	210	6	$210 \times 6 = 1260$
349	1650	$2 \times 3 \times 5^2 \times 11$	330	5	$330 \times 5 = 1650$
403	1980	$2^2 \times 3^2 \times 5 \times 11$	330	6	$330 \times 6 = 1980$
2887	18480	$2^4 \times 3 \times 5 \times 7 \times 11$	2310	8	$2310 \times 8 = 18480$
26357	210210	$2 \times 3 \times 5 \times 7^2 \times 11 \times 13$	30030	7	$30030 \times 7 = 210210$
467422	4594590	$2 \times 3^3 \times 5 \times 7 \times 11 \times 13 \times 17$	510510	9	$510510 \times 9 = 4594590$

Observation on primorial intersections

There’s a neat intersection between A396594 and numbers with a primorial kernel (A055932). For $k$ in that intersection, the kernel ratio just equals $\Omega(k)$:

$\frac{k}{\text{rad}(k)} = \Omega(k)$

Looking at the irregular triangle $T(n, j) = \text{A002110}(n) \times j$ (where A002110 gives the primorials $2, 6, 30, 210, 2310, \dots$), a solution exists exactly when $\Omega(T(n, j)) = j$. That alone generates arbitrarily large-$\Omega$ solutions:

• $(n=1, j=2) \implies 2 \times 2 = 4$
• $(n=2, j=3) \implies 6 \times 3 = 18$
• $(n=2, j=4) \implies 6 \times 4 = 24$
• $(n=4, j=5) \implies 210 \times 5 = 1050$
• $(n=4, j=6) \implies 210 \times 6 = 1260$
• $(n=5, j=8) \implies 2310 \times 8 = 18480$
• $(n=7, j=9) \implies 510510 \times 9 = 4594590$
• $(n=12, j=13) \implies 7420738134810 \times 13 = 96469595752530$

Checking $30,000$ terms of A396157 by computer, every non-conforming composite satisfies the strict inequality:

$\text{rad}(k) \times \Omega(k) < \text{rad}(k)^2 \implies \Omega(k) < \text{rad}(k)$

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3. Proofs of structural boundaries in A396594

Now for the boundary proofs — how A396594 relates to powerful numbers (A001694) and squarefree numbers (A005117).

Theorem 1 (the powerful number boundary)

$k = 4$ is the only powerful number in A396594. It follows that A396594 shares no elements with the Achilles numbers (A052486).

Proof by kernel scaling

1. If $k$ is powerful (A001694), every prime factor has exponent $\ge 2$, so the square of the radical divides $k$:

$\text{rad}(k)^2 \mid k \implies k \ge \text{rad}(k)^2 \implies \sqrt{k} \ge \text{rad}(k)$

2. Dividing $k \ge \text{rad}(k)^2$ by $\text{rad}(k)$ gives the baseline constraint for powerful numbers:

$\frac{k}{\text{rad}(k)} \ge \text{rad}(k)$

3. Membership in A396594 requires $k / \text{rad}(k) = \Omega(k)$. Plugging that in, any powerful member must satisfy:

$\Omega(k) \ge \text{rad}(k)$

4. To see whether that’s even possible, minimize $\text{rad}(k)$ for a given $h = \omega(k)$ by picking the primorial kernel $\text{rad}(k) = P_h = \text{A002110}(h)$.

• $h = 1$ ($\text{rad}(k) = 2$): these are prime powers $k = 2^m$. We need $2^m / 2 = \Omega(2^m)$, i.e. $2^{m-1} = m$. $m=1$: $2^0 = 1 \ne 0$, fails. $m=2$: $2^1 = 2$ — works, giving $k = 4 = 2^2$. For $m \ge 3$, exponential growth outpaces linear growth ($2^{m-1} > m$), so no further powers of 2 qualify.
• $h = 2$ ($\text{rad}(k) = 2 \times 3 = 6$): the smallest powerful number here is $k = 6^2 = 36$. $\text{rad}(36) = 6$, $\Omega(36) = 4$, so $36 / \text{rad}(36) = 6 > 4 = \Omega(36)$ — the quotient is already ahead. Multiplying by a prime $p \in \{2, 3\}$ bumps $\Omega(k)$ by just $1$ but scales the quotient by $p$. So the gap only grows:

$\frac{k \cdot p}{\text{rad}(k)} - \Omega(k \cdot p) = p\frac{k}{\text{rad}(k)} - (\Omega(k) + 1) > 0$

• $h \ge 3$ ($\text{rad}(k) \ge 30$): at $h=3$, the smallest powerful number is $30^2 = 900$, where $\text{rad}(900) = 30$ and $\Omega(900) = 6$ — already a $5\times$ gap. As $h$ grows, the primorial $\text{A002110}(h)$ grows super-exponentially ($e^{(1+o(1))h \ln h}$), while $\Omega(k)$ only grows logarithmically relative to $k$. The gap only widens further.

5. So $\text{rad}(k) > \Omega(k)$ for every powerful number except $k=4$. Since every Achilles number (A052486) is powerful and greater than 4 (the smallest is $72 = 2^3 \times 3^2$), none of them can appear in A396594. $\blacksquare$

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The Squarefree Boundary

The only squarefree numbers (A005117) in A396594 are the primes (A000040).

Proof by contradiction:

1. Let $k > 1$ be squarefree. No exponent exceeds 1, so $\text{rad}(k) = k$.
2. Every prime factor has multiplicity 1, so $\Omega(k) = \omega(k)$.
3. Suppose $k$ belongs to A396594, so it satisfies:

$k = \text{rad}(k) \times \Omega(k)$

4. Substitute $\text{rad}(k) = k$ and $\Omega(k) = \omega(k)$:

$k = k \times \omega(k) \implies \omega(k) = 1$

5. The only squarefree integers with $\omega(k) = 1$ are the primes themselves.
6. For any composite squarefree number (like $k = 6, 10, 14, 15, \dots$, catalogued in A120944), $\omega(k) \ge 2$, which gives:

$\text{rad}(k) \times \Omega(k) = k \times \omega(k) \ge 2k > k$

So no composite squarefree number can satisfy the identity. $\blacksquare$

Corollary 1 (squaremid subset)

Let $\mathcal{C}$ be the composite terms of A396594 excluding $4$. Since $\mathcal{C}$ has no squarefree numbers (Theorem 2) and no powerful numbers (Theorem 1), every term in $\mathcal{C}$ is both non-squarefree and non-powerful — meaning:

$\left( \text{A396594} \setminus \text{A175787} \right) \setminus \{4\} \subset \text{A332785}$

where A175787 is the primes plus $1$, and A332785 is the squaremid domain.

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4. Partitioning the “squaremid” universe (A332785)

A332785 collects the numbers $k$ that are neither squarefree nor powerful. In terms of the kernel ratio:

$\text{A332785} = \left\{ k \in \mathbb{Z}^+ : \text{rad}(k)^2 \nmid k \land \omega(k) < \Omega(k) \right\}$

We can split this domain into three pieces by intersecting with the radical-prime resonance functions.

4.1 Definition of the partition sets

• Block 1: $V_{7917}$ — The $\omega$-Resonant Squaremid Numbers (A397499): A332785 intersected with A397221 (where $k = \text{rad}(k) \times \omega(k)$):

$V_{7917} = \left\{ k \in \text{A332785} : \frac{k}{\text{rad}(k)} = \omega(k) \right\}$

First terms: $12, 20, 28, 44, 52, 68, 76, 90, 92, 116, 124, 126, 148, 164, 172, 188, 198, 212, \dots$

• Block 2: $V_{7918}$ — The $\Omega$-Resonant Squaremid Numbers (A397500): A332785 intersected with A396594 (where $k = \text{rad}(k) \times \Omega(k)$):

$V_{7918} = \left\{ k \in \text{A332785} : \frac{k}{\text{rad}(k)} = \Omega(k) \right\}$

First terms: $18, 24, 40, 45, 56, 63, 88, 99, 104, 117, 136, 152, 153, 171, 184, 207, 232, 248, \dots$

• Block 3: $V_{7919}$ — The Non-Resonant Squaremid Remainder (A396367): Everything left over — terms that resonate with neither prime-counting metric:

$V_{7919} = \text{A332785} \setminus \left( V_{7917} \cup V_{7918} \right)$

First terms: $48, 50, 54, 60, 75, 80, 84, 96, 98, 112, 120, 132, 135, 140, 147, 150, 156, 160, \dots$

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4.2 Theorem 3 (strict disjunction of resonant blocks)

$V_{7917}$ (A397499) and $V_{7918}$ (A397500) don’t overlap: $V_{7917} \cap V_{7918} = \emptyset$.

Proof by contradiction:

1. Suppose some $n \in \text{A332785}$ belongs to both $V_{7917}$ and $V_{7918}$.
2. Membership in $V_{7917}$ requires:

$\text{rad}(n) \times \omega(n) = n$

3. Membership in $V_{7918}$ requires:

$\text{rad}(n) \times \Omega(n) = n$

4. Setting these equal:

$\text{rad}(n) \times \omega(n) = \text{rad}(n) \times \Omega(n)$

5. Since $n \in \text{A332785}$ means $n \ge 12$, we know $\text{rad}(n) \ge 6 > 0$, so we can divide both sides by $\text{rad}(n)$:

$\omega(n) = \Omega(n)$

6. But $\omega(n) = \Omega(n)$ holds only when $n$ is squarefree (A005117).
7. A332785 excludes squarefree numbers entirely — every element has at least one prime factor with multiplicity $\ge 2$, so:

$\Omega(k) > \omega(k)$

8. That’s a direct contradiction with step 6. So no integer belongs to both sets. $\blacksquare$

Asymptotic density distribution

Checking the distribution of these blocks across the first $N = 2^{20} = 1,048,576$ integers shows a fairly lopsided split. Of the $409,043$ terms of A332785 below $2^{20}$:

• Block $V_{7917}$ (A397499): $39,958$ terms ($\approx 9.77\%$).
• Block $V_{7918}$ (A397500): $22,831$ terms ($\approx 5.58\%$).
• Block $V_{7919}$ (A396367): $346,254$ terms ($\approx 84.65\%$).

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5. Generalized $f$-resonance and the naked prime constraint

To avoid this framework collapsing into a set of one-off tricks for isolated sequences, it’s worth generalizing beyond specific prime-counting functions to the broader tension between prime multiplicities and arbitrary arithmetic functions.

Let $f(k)$ be any arithmetic function with polynomial growth relative to $\Omega(k)$ — that is, some constant $c \ge 1$ with:

$f(k) \le \Omega(k)^c$

Call an integer $k > 1$ $f$-resonant if:

$k = \text{rad}(k) \times f(k) \iff \frac{k}{\text{rad}(k)} = f(k)$

This covers A397221 ($f = \omega$), A396594 ($f = \Omega$), and opens the door to similar sequences built on the divisor function $d(k)$ (A000005) or sum of prime factors $\text{sopf}(k)$ (A001414).

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5.1 Theorem 4 (the naked prime constraint for greatest prime factors)

For $k \in V_{7917}$ (A397499), the greatest prime factor $g = \text{gpf}(k)$ (A006530) is always “naked” — its multiplicity is exactly $1$ ($g^2 \nmid k$).

Proof by modular divisibility

1. Let $k \in V_{7917}$ have factorization $k = p_1^{a_1} \cdots p_m^{a_m}$ with $p_1 < p_2 < \dots < p_m$. So $g = p_m$.
2. Suppose $g$ is not naked, i.e. $a_m \ge 2$, so $g^2 \mid k$.
3. The kernel quotient then carries an exponent of $a_m - 1 \ge 1$ on $g$:

$\frac{k}{\text{rad}(k)} = p_1^{a_1 - 1} \cdots g^{a_m - 1}$

4. So $g$ must divide the kernel ratio:

$g \ \Big| \ \frac{k}{\text{rad}(k)}$

5. By definition of $V_{7917}$, that kernel ratio equals $\omega(k) = m$. So:

$g \mid m \implies g \le m$

6. But the $m$-th prime is always strictly greater than its index $m$ (e.g. $p_1 = 2 > 1$, $p_2 = 3 > 2$, $p_3 = 5 > 3$), so:

$g = p_m > m = \omega(k)$

7. A prime can’t divide a smaller positive integer, so $g \mid \omega(k)$ is impossible.
8. So $a_m \ge 2$ can’t hold — $g$ must have multiplicity 1, making $g^2 \nmid k$ true across all of Block 1. $\blacksquare$

(For Block 2, $V_{7918}$ / A397500, this holds for every composite term except $k = 18 = 2 \times 3^2$, where $\text{gpf}(18) = 3$ divides $\Omega(18) = 3$. For $k > 18$ it holds without exception.)

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5.2 Theorem 5 (logarithmic upper bounds on prime multiplicities)

If $k = \prod p_i^{a_i}$ is $f$-resonant under a polynomial-growth function $f(k) \le \Omega(k)^c$, then no prime exponent $a_p$ can exceed:

$a_p \le 1 + c \log_2(\Omega(k))$

Proof:

1. By $f$-resonance, the kernel quotient equals $f(k)$:

$\prod_{p \mid k} p^{a_p - 1} = f(k)$

2. And $f(k) \le \Omega(k)^c$, so:

$\prod_{p \mid k} p^{a_p - 1} \le \Omega(k)^c$

3. Every factor $p^{a_p - 1}$ in that product is $\ge 1$ (since $p \ge 2$), so no single term can exceed the whole product:

$p^{a_p - 1} \le \prod_{i=1}^{m} p_i^{a_i - 1} \le \Omega(k)^c$

4. Take $\log_2$ of both sides of $p^{a_p - 1} \le \Omega(k)^c$:

$(a_p - 1) \log_2(p) \le c \log_2(\Omega(k))$

5. Since the smallest prime is $2$, $\log_2(p) \ge 1$ always, so:

$(a_p - 1)(1) \le (a_p - 1) \log_2(p) \le c \log_2(\Omega(k))$

6. Isolating $a_p$:

$a_p \le 1 + c \log_2(\Omega(k)) \ \ \blacksquare$

Specialization to A396594 and A397221

For our two sequences (A396594 with $f = \Omega$, A397221 with $f = \omega$), $c = 1$ exactly. So the hard limit on any prime multiplicity in these sequences is:

$a_p \le 1 + \log_2(\Omega(k))$

Which explains why an exponent of $a_p = 4$ only shows up once $\Omega(k) \ge 2^{4-1} = 8$ — that’s why high exponents only appear in large terms like $k = 18480$ ($\Omega = 8$, containing $2^4$).

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5.3 Theorem 6 (the asymptotic naked prime law)

In any $f$-resonant integer $k$ with $f(k) \le \Omega(k)^c$, a prime factor $p$ larger than $\Omega(k)^c$ is forced to have multiplicity exactly $1$.

Proof by contradiction

1. Suppose some prime $p \mid k$ has $p > \Omega(k)^c$ but isn’t naked — i.e. $a_p \ge 2$.
2. Then $a_p - 1 \ge 1$.
3. Since $p$ is an integer $\ge 2$, raising it to a power $\ge 1$ gives at least $p$ itself:

$p^{a_p - 1} \ge p^1 = p$

4. Combined with $p > \Omega(k)^c$:

$p^{a_p - 1} \ge p > \Omega(k)^c \implies p^{a_p - 1} > \Omega(k)^c$

5. But Theorem 5 (step 3) already showed:

$p^{a_p - 1} \le \prod_{i=1}^{m} p_i^{a_i - 1} = f(k) \le \Omega(k)^c$

6. That’s a contradiction — $\Omega(k)^c < p^{a_p - 1} \le \Omega(k)^c$ can’t both be true.
7. So $a_p \ge 2$ is impossible. Any prime factor above $\Omega(k)^c$ must have exponent exactly $1$. $\blacksquare$

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6. Structural resonance when $\omega(k)$ is prime

Last piece: how rigid does Block 1 ($V_{7917}$ / A397499) get when $\omega(k)$ itself is a prime (A000040)?

Theorem 7 (prime-index rigidity)

Let $k \in V_{7917}$ with $\omega(k) = q$, $q$ prime. Then $k$ has exactly one non-squarefree prime factor — and it has to be $q$ itself, with multiplicity exactly $2$ ($a_q = 2$). Every other one of the $\omega(k) - 1$ prime factors is linear ($a_i = 1$).

Proof:

1. By definition of $V_{7917}$, the kernel ratio equals the distinct prime count:

$\frac{k}{\text{rad}(k)} = \prod_{i=1}^{m} p_i^{a_i - 1} = \omega(k) = q$

2. Since $q$ is prime, its only divisor greater than 1 is itself. So the product $\prod p_i^{a_i - 1}$ has to equal $q^1$.
3. By unique factorization, that only happens if:

• exactly one term has base $p_j = q$ with exponent $a_j - 1 = 1$, i.e. $a_j = 2$,
• every other term has exponent $a_i - 1 = 0$, i.e. $a_i = 1$.

4. So $k$‘s factorization is forced into this rigid shape:

$k = q^2 \times \prod_{\substack{i=1 \\ p_i \ne q}}^{q} p_i$

with exactly $q$ distinct prime factors total. $\blacksquare$

Empirical verification

This shows up cleanly in the early prime-index terms of Block 1 (A397499):

• $k = 12$: $\omega(12) = \omega(2^2 \times 3) = 2$ (prime). The repeated prime is $2$, with exponent $2$.
• $k = 20$: $\omega(20) = \omega(2^2 \times 5) = 2$ (prime). Repeated prime is $2$, exponent $2$.
• $k = 28$: $\omega(28) = \omega(2^2 \times 7) = 2$ (prime). Repeated prime is $2$, exponent $2$.
• $k = 315$: $\omega(315) = \omega(3^2 \times 5 \times 7) = 3$ (prime). Repeated prime is $3$, exponent $2$, and the other two ($5$ and $7$) are naked.

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7. Conclusion and future directions

Formalizing A396594 and its overlap with the squaremid universe A332785 shows that radical-prime resonance identities put fairly tight constraints on prime factorizations.

Powerful numbers (A001694) and squarefree numbers (A005117) act as hard outer boundaries, which lets us isolate the composite core into three partition blocks (A397499, A397500, A396367). The generalization to $f$-resonant integers gives logarithmic growth limits and the asymptotic naked prime law, though there’s likely more structure here than we’ve captured.

Worth chasing next:

1. How the density ratio $|V_{7917}(X)| / |V_{7918}(X)|$ behaves asymptotically as $X \to \infty$.
2. Whether an upper bound exists for the least prime factor $\text{lpf}(k)$ among non-conforming composite trajectories (A396157).
3. What resonance sequences built from non-additive functions look like — Euler’s totient $\phi(k)$ (A000010) and Dedekind’s psi function $\psi(k)$ (A001615) seem like natural next candidates.

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References and OEIS links

• A000010: Euler’s totient function $\phi(n)$.
• A000040: The prime numbers.
• A001221: Number of distinct prime factors $\omega(n)$.
• A001222: Total number of prime factors $\Omega(n)$.
• A001597: Perfect powers $k^m$ ($m > 1$).
• A001694: Powerful (or squareful) numbers.
• A002110: Primorial numbers (product of first $n$ primes).
• A003557: The kernel quotient $n / \text{rad}(n)$.
• A005117: Squarefree numbers.
• A006530: Greatest prime factor $\text{gpf}(n)$.
• A007947: Squarefree kernel $\text{rad}(n)$.
• A020639: Least prime factor $\text{lpf}(n)$.
• A052486: Achilles numbers (powerful but not perfect powers).
• A055932: Numbers with a primorial squarefree kernel.
• A096156: Numbers of the form $p^2 q$ with primes $p < q$.
• A120944: Composite squarefree numbers.
• A175787: Prime numbers and $1$.
• A332785: “Squaremid” numbers (neither squarefree nor powerful).
• A366825: Numbers where canonical prime exponents are setwise coprime with at least one unity.
• A396157: Non-conforming composite terms of A396594 ($S_{20260704}$).
• A396367: The non-resonant squaremid remainder ($V_{7919}$).
• A396594: Numbers satisfying $k = \text{rad}(k) \times \Omega(k)$.
• A397221: Numbers satisfying $k = \text{rad}(k) \times \omega(k)$.
• A397499: The $\omega$-resonant squaremid partition block ($V_{7917}$).
• A397500: The $\Omega$-resonant squaremid partition block ($V_{7918}$).