Notes on A376936. Michael Thomas De Vlieger, Saint Louis, Missouri, 202410181030. Abstract. --------- We present theorems associated with the sequence A376936, numbers n that have coreful divisors d, n/d such that neither d | n/d nor n/d | d. We show that A376936 ⊂ A286708 ⊂ A1694, and further, that A376936 is the list of powerful numbers that are not prime powers, divisible by at least 2 prime cubes. Introduction. ------------- The term "coreful" has the meaning that 2 or more numbers have the same squarefree kernel ϰ. * We employ the term "codivisor" to mean complementary divisors d | n and n/d, since d × n/d = n. In other words, if d is a divisor of n, then n/d is its codivisor, and {d, n/d} is a pair of codivisors. Let property Q be the existence of "coreful" codivisors {d, n/d} such that rad(d) = rad(n/d) but neither d nor n/d divides the other, where rad = A7947. For concision, we employ the phrase "coreful divisors absent divisibility". The number of distinct prime factors is rendered by "little omega" ω = A1221. "Big omega" Ω is the number of prime factors with multiplicity, A1222. Thoughts. --------- Tautologies: 1. The number of distinct prime factors is conserved among d, n/d, and n. 2. Ω(n) = Ω(d) + Ω(n/d). Lemma A1. Coreful divisors absent divisibility are dissimilar, since equality implies divisibility. Lemma A2. rad(d) = rad(n/d) = ϰ implies rad(n) = ϰ. Therefore {d, n/d, n} are coreful. (ϰ is script kappa). Proof. Suppose not. Suppose that n has a prime factor q that divides neither d nor n/d. Then d × n/d is also not divisible by q, contradicting the definition of n/d. Lemma A3. Only nonsquarefree n ∈ A013929 posess property Q. Proof. Divisors d of squarefree n are also squarefree. These d are products of subsets of the prime factors of n, and the sets of prime factors of codivisors {d, n/d} are complementary. Lemma A4. Only n ∈ A024619 posess property Q. Proof. Divisors of prime powers p^k, k >= 0, are prime powers p^j, j = 0..k. Even k imply sqrt(p^k) = s an integer, hence a divisor whose codivisor is itself and it is clear that s | s. For all other codivisor pairs, since all are dissimilar powers of the same prime, the smaller of the pair divides the other. Thus there are no codivisor pairs that do not have a divisor relationship. Lemma A5. Trivial divisors are not coreful codivisors absent divisibility, since 1 | n and n | n. Theorem A. A376936 is contained in A126706, where the latter sequence is the intersection of A024619 and A013929. Consequence of Lemmas A3 and A4. Corollary A6. Coreful divisors absent divisibility are in A024619. Consequence of Lemmas A2 and A4. Corollary B1. Coreful codivisors are dissimilar and both in A126706. Corollary B2. Coreful codivisors occur in S_ϰ(k) for k > 1, where S_ϰ = { ϰ × m : rad(m) | ϰ } and ϰ = rad(d) = rad(n/d). This, since S_ϰ(1) = ϰ is the only squarefree term. Corollary B3. Let d = S_ϰ(i) and n/d = S_ϰ(j) such that 1 < i < j. Then n = S_ϰ(k) for k > j. Consequence of Lemma A2 and Corollary B2, since d < n/d < n by construction in this corollary. Example: 12 × 18 = 216; squarefree kernel = 6, hence all are in S₆ = A033845. A033845(2) = 12, A033845(3) 18, and A033845(14) = 216. Lemma B4. Coreful divisors absent divisibility are in A013929. Proof. Suppose squarefree d | n. From Lemma A2, we know rad(d) = rad(n/d) = rad(n), therefore d = rad(n/d) and thus d | n/d, contradicting d as absent divisibility. We may apply same logic to n/d and obtain the same conclusion that n/d | d. Theorem B. Coreful divisors absent divisibility are in A126706. Consequence of Corollary A6 and Lemma B4. Corollary B5. The smallest possible coreful divisor d | m is S_rad(m)(2) = lpf(m) × rad(m), a number M ∈ A366825. Consequence of Corollary B2. For squarefree composite ϰ ∈ A120944, number S_ϰ(2) ∈ A366825 is the smallest term in S_ϰ that is nonsquarefree; A366825 ⊂ A126706. It has a particular construction: S_ϰ(2) = lpf(ϰ) × ϰ = A020736(A120944(i)) × A120944(i) = p × A056911(i), gcd(p, A056911(i)) = 1, p < lpf(A056911(i)). This is a "minimally tantus" number d [1] and is the smallest possible coreful divisor of m. Theorem C. A376936 ⊂ A286708, where A286708 contains powerful numbers that are not prime powers. Proof. Suppose n indivisible by rad(n)², where prime factor p | n but p² does not divide n. Then p | d implies n/d not divisible by p, contradicting the definition of coreful codivisors. We know that 36 is not in A376936 since for rad(36) = 6, S₆(2) × S₆(3) = 12 × 18 are the smallest coreful divisors absent divisibility of a 3-smooth number. Therefore A376936 is a proper subset of A286708. The latter is the intersection of A1694 and A126706. Theorem D. A162142 ⊂ A376936, where A162142 contains cubes of squarefree semiprimes A6881. Consequence of Corollaries B2, B3, Corollary B5, and the following: Proof. Let n ∈ A162142, hence n = (pq)³ with ω(n) = 2. Then rad(n) is a squarefree semiprime p × q ∈ A6881. We employ a minimally tantus number M = lpf(n) × rad(n). It is clear that codivisors p²q × pq² = p³q³ = (pq)³, and neither divide the other. Hence we demonstrate that A162142 ⊂ A376936. Theorem E. A177493 ⊂ A376936, where A177493 contains cubes of squarefree composites A120944. Consequence of Corollaries B2, B3, B5, and the following: Proof: Building on the method used to find coreful codivisors of n ∈ A162142, we attempt the following: Let M = lpf(n) × rad(n). Set d = M and n/d = n/m. Then we have p²q...r × pq²...r² = p³q³...r³ = (pq..r)³, and neither divide the other. Therefore we show that A177493 ⊂ A376936. Theorem F. A372695 ⊂ A376936, where A372695 contains cubefull numbers that are not prime powers. Consequence of Corollaries B2, B3, B5, and following logic in Theorems C and D. Corollary G1. The number of prime factors with multiplicity of coreful codivisors exceeds the number of distinct prime factors. This is to say, for ω(d) = ω(d/n) = ω(n) = W, Ω(d) > W, Ω(n/d) > W. (See Theorems C-F) Corollary G2. Number of prime factors with repetition of n is at least twice the number of distinct prime factors of n, i.e., Ω(n) ≥ 2 × ω(n). Lemma G3. For W = ω(n) = 2, n = p^i × q^j, primes p < q, with both i ≥ 3 and j ≥ 3. Proof. Theorem C shows that n is powerful (squareful) and in A126706, therefore i and j must exceed 1. Theorem B shows that coreful divisors absent divisibility are in A126706, while Lemma A2 shows that rad(d) = rad(n/d) = rad(n). Suppose i = j = 2. The divisor d = p²q leaves n/d = q, while d = pq² leaves n/d = p, and W is not conserved, violating tautology 1, and we have no solution that satisfies the theorems. Therefore squares of squarefree semiprimes n ∈ A085986 are not in A376936. Suppose i < j, i = 2, j = 3, hence Ω(n) = 5. It is clear that, given W = 2, we cannot produce coreful divisors absent divisibility with d such that Ω(d) = 3 and Ω(n/d) = 2, since n/d would be squarefree and thus divide d. Therefore both i and j must both be at least 3. Finally, setting i < j, i = 3, hence Ω(n) > 2 × W, we can always obtain coreful divisors absent divisibility, since we at least have 1 pair (p² × q) × (p × q^(j−1)). Increasing i has no effect on the existence of the codivisors (p² × q) × (p × q^(j−1)). Theorem G. For W = ω(n) > 2, n is a powerful product of at least 2 prime cubes. This is to say that all prime power factor exponents exceed 1, but 2 such must exceed 2. Proof. Building on Lemma G3, we have the codivisor pair (p² × q × rad(n)/pq) × (p × q^(j−1) × rad(n)/pq). Corollary G4. Prime signatures in this sequence include {3+, 3+}, {3+, 3+, 2+}, and {3+, 3+, 2+, ..., 2+}. Examples of the first form: {216, 432, 648, 864, 1000}. Examples of the second form: {5400, 9000, 10584, 10800, 13500}. Examples of the third form: {264600, 441000, 529200, 617400, 653400}. Theorem H. A162142, A177493, and A372695 are proper subsets of A376936. Proof. Argument by construction: the number 5400 = 2³ × 3³ × 5² is not in any of A162142, A177493, and A372695, but has coreful divisors absent divisibility, i.e., 60 × 90 = (2² × 3 × 5) × (2 × 3² × 5) = 2³ × 3³ × 5² = 5400. Conclusion. ----------- The sequence A376936 contains numbers that are powerful but not prime powers, divisible by at least 2 distinct prime cubes. Appendix: ========= Notes. ------ * In personal writings the term "coregular" is used instead of "coreful", since n-regular k refers to k and n such that rad(k) | n, and n-coregular k refers to k and n such that rad(k) = rad(n). References. ----------- [1] Michael De Vlieger, Minimally Tantus Numbers (OEIS A366825), Simple Seq. Analysis SA20231220 (2023), DOI: 10.13140/RG.2.2.17929.42084. [2] Neil J. A. Sloane, The Online Encyclopedia of Integer Sequences, retrieved October 2024. Concerns sequences: A1221, A1222, A7947, A030078, A085986, A126706, A162142, A177493, A286708, A307958, A361430, A370329, A372695, A376514. Dedication. ----------- This work is dedicated to Prof. Olivier Gérard, fellow Editor in Chief of OEIS, who in current understanding passed away suddenly September 2024 on the island of Mayotte.