Alladi, K.; Erdős, Paul On an additive arithmetic function. (English) Zbl 0359.10038 Pac. J. Math. 71, 275-294 (1977). Let \(n\) be a positive integer, \(n=\prod\limits_{i=1}^rp_i^{\alpha_i}\) in canonical form, and let \(A(n)=\sum\limits_{i=1}^r\alpha_ip_i\). Clearly \(A\) is an additive arithmetic function. Assume the primes \(p_i\) are arranged so that \(p_1\leq p_2<\dots<p_r\). Define \(P_1(n)=p_r\) and, in general, \(P_k(n)=P_1(n/P_1(n)\dots\dots P_{k-1}(n))\) for \(k\leq\sum\limits_{i=1}^r\alpha_i\) and \(P_k(n)=0\) for \(k>\sum\limits_{i=1}^r\alpha_i\). If \(f\) and \(g\) are arithmetic functions such that \(\sum\limits_{n\leq x}f(n)\sim\sum\limits_{n<x}g(n)\), and if \(g\) is a well behaved function (e.g. polynomial, exponential), then \(g\) is referred to as the average order of \(f\). It is proved that for all positive integers \(m\) we have \[ \sum\limits_{n\leq x}P_m(n)\sim\sum\limits_{n\leq x}\{A(n)-P_1(n)-\dots-P_{m-1}(n)\}\sim k_mx^{1=(1/m)}/(\log x)^m \] where \(k_m\) is a positive constant depending only on \(m\). It follows almost immediately from this theorem that the average order of \(A(n)\) is \(\pi^2n/6\log n\). Let \(A^*(n)=\sum\limits_{i=1}^r\). Then the average order of \(A*(n)\) is also \(\pi^2n/6\log n\), and the average order of \(A(n)-A^*(n)\) is \(\log\log n\). For any fixed positive integer \(M\), the set of solutions to \(A(n)-A^*(n)=M\) has a positive natural density. Now \(A(n)=n\) if and only if \(n\) is a prime or \(n=4\). Call \(n\) a \`\` special number” if \(n\equiv0(\mod A(n))\) and \(n\neq A(n)\), and let \(\{\ell_n\}\) be the sequence of special numbers. This paper’s first author has previously proved that the sequence \(\{\ell_n\}\) is infinite [Srinivasa Ramanujan Commemoration Volume, Oxford Press, Madras, India, (1974) part II]. Denote by \(\mathcal L(x)\) the number of \(\ell_n\leq x\). It is shown that there exist positive constants \(c,c^{'}\) such that \[ \mathcal L(x)=0(xe^{-c\sqrt{\log x\log\log x}})\text{ and }\mathcal L\gg xe^{-c^{'}\sqrt{\log x\log\log x}} \] Finally, let \(\alpha(n)=(-1)^{A_{(n)}}\). It is proved that there exists a positive constant \(c^{''}\) such that \[ \sum\limits_{1\leq n\leq x}\alpha(n)=0(xe^{-c^{''}\sqrt{\log x\log\log x}}), \] and that \(\sum\limits_{n=1}^\infty\alpha(n)/n=0\). Reviewer: Betty Garrison (San Diego) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 5 ReviewsCited in 27 Documents MSC: 11N37 Asymptotic results on arithmetic functions 11K65 Arithmetic functions in probabilistic number theory PDFBibTeX XMLCite \textit{K. Alladi} and \textit{P. Erdős}, Pac. J. Math. 71, 275--294 (1977; Zbl 0359.10038) Full Text: DOI Online Encyclopedia of Integer Sequences: Number of partitions of n into prime parts. Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n). Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1. a(n) = sopfr(n!), where sopfr = A001414 is the integer log. Numbers k such that k / sopfr(k) is an integer, where sopfr = sum-of-prime-factors, A001414. Composite numbers that are divisible by the sum of their prime factors (counted with multiplicity). Partial sums of A006530. Dividend associated with A116536. Intersection of A046346 (numbers that are divisible by the sum of their prime factors, counted with multiplicity) and A097889 (numbers that are products of at least two consecutive primes). a(n) is the smallest m such that A001414(m)=n and ((m=0) mod n) and m/n is both squarefree and prime to n, or 0 if no such m exists. a(n) = sopfr(n) - sopf(n). Composite numbers n of which the sum of prime divisors of n (with repetition) equals the concatenation of two integers k and k + 1. a(n) is the number of odd values minus the number of even values of the integer log of all positive integers up to and including n. Numerators of the fractions f(n) such that (6/Pi^2)*f(n) is the asymptotic density of the numbers k with A280292(k) = sopfr(k) - sopf(k) = n.