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.
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, 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, 2, 3, 39, 0, 2, 0, 2, 3, 2, 7, 35, 0, 2, 0, 2, 0, 23, 5
Offset: 1
Examples
Fractions begin with 0, 0, 0, 0, 0, 2/3, 0, 0, 0, 2/5, 0, 2/3, ...
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Paul Erdős and Jean-Louis Nicolas, Grandes valeurs de fonctions liées aux diviseurs premiers consécutifs d'un entier, 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; alternative link.
Programs
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Mathematica
a[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, Numerator[Total[Most[p]/Rest[p]]]]; Array[a, 100] -
PARI
a(n) = {my(p = factor(n)[,1]); numerator(sum(i = 1, #p-1, p[i]/p[i+1]));}
Formula
a(n) = 0 if and only if n is a power of a prime (A000961).
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.