Erdos 1990 distribution values certain class arithmetic functions

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lemma_2: Schinzel's lemma, printed with proof by Erdős and Ivić, that the number of distinct prime factors of the product of the partition numbers P(1) up to P(n) tends to infinity with n; the proof gives no rate.

theorem_1: Erdős and Ivić's asymptotic formula, for two functions of the class B of nonnegative integer-valued functions with squarefull kernel and growth at most n^epsilon, for the count of n up to x with f(n) = g(n+1), with the constant A an explicit series over coprime squarefull pairs.

theorem_2: Erdős and Ivić's asymptotic formula, for two functions of the class B of nonnegative integer-valued functions with squarefull kernel and growth at most n^epsilon, for the sum of f(n)g(n+1) over n up to x, with the constant C an explicit series over coprime squarefull pairs.

theorem_3: Erdős and Ivić's inequality that, for all large x, the number D(x) of integers up to x that are values of the Abelian-group count a(m) is at least a third of C(x) log log x, where C(x) counts the distinct values of a(n) for n up to x.

theorem_4: Erdős and Ivić's lower bound that the number C(x) of distinct values of the Abelian-group count a(n) for n up to x exceeds every fixed power of log x once x is large, deduced from Schinzel's Lemma 2.

theorem_5: Erdős and Ivić's conditional theorem that the conjectured bound (4.13) on the sequence k_t, built from the first appearances of new prime factors of the partition numbers, would give C(x) = exp((log x)^{1/2+o(1)}) and D(x) = exp((log x)^{2/3+o(1)}) for the Abelian-group count.

theorem_6: Erdős and Ivić's theorem that at least x^{1/2} integers n in [x,2x] have the Abelian-group counts a(n+1), ..., a(n+k) all equal, for k the integer part of log x log log log x over 40 (log log x)^2.

theorem_7: Erdős and Ivić's theorem that, for a suitable C > 0, at least x^{1/2} integers n in [x,2x] have the Abelian-group counts a(n+1), ..., a(n+t) pairwise distinct, for t the integer part of C (log x/log log x)^{1/2}.

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Erdős, Paul and Ivić, Aleksandar, The distribution of values of a certain class of arithmetic functions at consecutive integers. Number Theory (Budapest, 1987), Colloq. Math. Soc. János Bolyai 51, North-Holland, Amsterdam (1990), 45--91. No notice is printed in the file, and no Crossref license is recorded; the hosting archive's site footer speaks for the site, not the paper (https://users.renyi.hu/~p_erdos/, read 2026-10-02, prints "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only."), and no publisher page for this edition was found, so none was consulted; the term is unstated.

Erdős and Ivić study s-functions, the nonnegative integer-valued arithmetic functions with f(n)=f(s(n)) for the squarefull part s(n) of n (p. 46), motivated by a(n), the number of non-isomorphic Abelian groups of order n. Section 2 works in the class B of s-functions with f(n) << n^ε for any ε > 0 (definition, p. 51). For f,g in B, Theorems 1 and 2 (pp. 52--53) give #{n ≤ x : f(n)=g(n+1)} = Ax + O(x^{3/4} log^4 x) and Σ_{n≤x} f(n)g(n+1) = Cx + O(x^{3/4+ε}), with A and C explicit series over pairs of coprime squarefull numbers weighted by Euler products; the proofs rest on Lemma 1 (pp. 53--57), a count of the solutions of ka - lb = 1 in squarefree k, l. Section 3 draws applications, such as (3.4), the count of n ≤ x with a(n)=a(n+1). Section 4 studies C(x), the number of distinct values of a(n) for n ≤ x, and D(x), the number of n ≤ x with n = a(m) for some m. Theorem 3 (p. 65) gives D(x) ≥ (1/3)C(x) log log x for x ≥ x_0; Theorem 4 (p. 71) gives C(x) > (log x)^A for every fixed A > 0 and x ≥ x_0(A); and Theorem 5 (p. 73) shows that the hypothesis (4.13), k_t << t^{1+ε}, would give C(x) = exp((log x)^{1/2+o(1)}) and D(x) = exp((log x)^{2/3+o(1)}). Here k_t is the sequence of p. 70, integers 2 = k_1 < ... < k_t with primes 2 = q_1 < ... < q_t such that q_j divides P(k_j) but no earlier P(k_l), P the partition function. Section 5 constructs long blocks: Theorem 6 (p. 76) gives at least x^{1/2} integers n in [x,2x] with a(n+1)=...=a(n+k) for k = [log x log log log x/(40(log log x)^2)], and Theorem 7 (p. 76) gives at least x^{1/2} integers n in [x,2x] with a(n+1),...,a(n+t) all distinct for t = [C(log x/log log x)^{1/2}], for a suitable C > 0; both plant prescribed squarefull parts by the Chinese remainder theorem.

The input for Erdős Problem 1106 is Lemma 2 (p. 69), an unpublished result of A. Schinzel that the paper proves on pp. 69--70: the number of distinct prime factors of P(1)P(2)...P(n) tends to infinity with n. The proof supposes that finitely many primes q_1,...,q_r account for the prime factors of every P(n), n ≥ 2, and reaches a contradiction from Tijdeman's theorem on close integers composed of a fixed finite set of primes together with the asymptotic formula (4.9) for P(n), which the paper cites to Knopp's book. The lemma gives no rate.

Source: https://users.renyi.hu/~p_erdos/Erdos.html.

Read status. Claims checked: Lemma 2 and Theorems 1 to 7, with the definitions they use, were read clause by clause on the printed pages. The proofs were read for their structure, not checked step by step.

Bears on. #1106: Lemma 2 states that the number of distinct prime factors of p(1)p(2)...p(n) tends to infinity, the problem's first question, with no rate; the paper does not address the second question, whether that number exceeds n for all large n.

Results. Lemma 2 (p. 69, A. Schinzel); Theorem 1 (pp. 52--53, with the definition of B, p. 51); Theorem 2 (p. 53); Theorem 3 (p. 65); Theorem 4 (p. 71); Theorem 5 (p. 73, with the hypothesis (4.13), p. 71); Theorem 6 (p. 76); Theorem 7 (p. 76). Lemma 1 (pp. 53--54) is the counting step of Theorems 1 and 2, summarized on their pages.

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