Dekoninck 2011 distance between smooth numbers

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De Koninck, Jean-Marie and Doyon, Nicolas, On the distance between smooth numbers. Integers 11 (2011), #A25, 22 pp. The file prints no license line; the journal's site states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License so that all content is freely available without charge to the users or their institutions." (https://math.colgate.edu/~integers/, read 2026-10-02).

For n >= 2 the authors set delta(n) to be the distance from n to the nearest integer m != n, m >= 1, whose largest prime factor P(m) is at most P(n), calling delta(n) the index of isolation; by equation (1) on p. 2, delta(2^a)=2^{a-1}, and the paper notes that delta(n) <= n/2 for all n >= 2. A heuristic argument (Theorem 5, p. 7, conditional on their Hypothesis A) gives sum_{n<=x} 1/delta(n) = (4 log 2 - 2 + o(1)) x as x tends to infinity. For the generalized quantity delta_f(n) = min over m >= 1, m != n, with f(m) <= f(n) of |n-m| attached to an arbitrary real-valued arithmetic function f (with delta_f(1)=1), they prove unconditionally in Theorem 10 (p. 12) that sum_{a <= n < b} 1/delta_f(n) >= 2(b-a)/3 - 2/3 for positive integers a<b (the abstract states the bound without the -2/3), and in Theorem 14 (p. 17) that if f(m) >= f(a) for all m in [a,b) then sum_{a<n<b} delta_f(n) <= (b-a) log(b-a)/(2 log 2). Lemma 3 (p. 6) shows that for fixed B >= 3 there is c=c(B)>0 with delta(n) > n/(log n)^c for all B-smooth n large enough, confirming that genuinely smooth numbers are very isolated, although small examples such as n = 11859211 with delta(n)=1 show the rule of thumb fails for small n. The work bears on problem 372, which asks for infinitely many n with P(n)>P(n+1)>P(n+2): Section 3 (p. 6) recalls Balog's 2001 bound, that the number of n <= x with P(n-1)>P(n)>P(n+1) is >> sqrt(x), and Balog's expectation that such n have density 1/6, and on p. 7 states Hypothesis A, the unproved assumption that each ordering of P(n), ..., P(n+k-1) has probability 1/k!. The paper proves nothing new about Problem 372.

Source: https://math.colgate.edu/~integers/vol11.html.

Bears on. #372

Results to transcribe.

• Heuristic (abstract; Theorem 5, assuming Hypothesis A): A heuristic argument gives sum_{n<=x} 1/delta(n) = (4 log 2 - 2 + o(1))x.
• Theorem 10 (p. 12), lower bound for delta_f: For positive integers a<b, sum_{a <= n < b} 1/delta_f(n) >= 2(b-a)/3 - 2/3.
• Theorem 14 (p. 17), upper bound for delta_f: If f(m) >= f(a) for all m in [a,b), then sum_{a<n<b} delta_f(n) <= (b-a)log(b-a)/(2 log 2).
• Lemma 3 (p. 6): For fixed B >= 3 there is c(B)>0 and n_0(B) with delta(n) > n/(log n)^c for all B-smooth n >= n_0.
• Equation (1) (p. 2): delta(2^a) = 2^{a-1} for every a >= 1, so the most isolated integer up to x is the largest power of 2 not exceeding x.