cp's OEIS Frontend

A381638 Numerators of Sum_{i=1..omega(n)-1} p_{i}/p_{i+1}, where omega(n) = A001221(n) and p_1 < p_2 < ... p_omega(n) are the distinct prime factors of n; a(1) = 0.

%I A381638 #8 Mar 03 2025 13:33:26
%S A381638 0,0,0,0,0,2,0,0,0,2,0,2,0,2,3,0,0,2,0,2,3,2,0,2,0,2,0,2,0,19,0,0,3,2,
%T A381638 5,2,0,2,3,2,0,23,0,2,3,2,0,2,0,2,3,2,0,2,5,2,3,2,0,19,0,2,3,0,5,31,0,
%U A381638 2,3,39,0,2,0,2,3,2,7,35,0,2,0,2,0,23,5
%N A381638 Numerators of Sum_{i=1..omega(n)-1} p_{i}/p_{i+1}, where omega(n) = A001221(n) and p_1 < p_2 < ... p_omega(n) are the distinct prime factors of n; a(1) = 0.
%H A381638 Amiram Eldar, <a href="/A381638/b381638.txt">Table of n, a(n) for n = 1..10000</a>
%H A381638 Paul Erdős and Jean-Louis Nicolas, <a href="https://doi.org/10.1515/9783110852790.169">Grandes valeurs de fonctions liées aux diviseurs premiers consécutifs d'un entier</a>, in: Jean-Marie de Koninck and Claude Levesque (eds.), Théorie des nombres / Number Theory, Proceedings of the International Number Theory Conference held at Université Laval, July 5-18, 1987, De Gruyter, 1989; <a href="https://users.renyi.hu/~p_erdos/1989-08.pdf">alternative link</a>.
%F A381638 a(n) = 0 if and only if n is a power of a prime (A000961).
%F A381638 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{q prime} (1/q^2) * Sum_{primes p < q} Product_{primes r, p < r < q} (1-1/r). This sum converges slowly: for primes q that are not exceeding 10^9, 10^10, 10^11, and 10^12, the sums are 0.5399..., 0.5447..., 0.5487..., and 0.5520..., respectively.
%e A381638 Fractions begin with 0, 0, 0, 0, 0, 2/3, 0, 0, 0, 2/5, 0, 2/3, ...
%t A381638 a[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, Numerator[Total[Most[p]/Rest[p]]]]; Array[a, 100]
%o A381638 (PARI) a(n) = {my(p = factor(n)[,1]); numerator(sum(i = 1, #p-1, p[i]/p[i+1]));}
%Y A381638 Cf. A000961, A001221, A381639 (denominators), A381640, A381641.
%K A381638 nonn,easy,frac
%O A381638 1,6
%A A381638 _Amiram Eldar_, Mar 03 2025