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A381641 Numerators of Sum_{i=1..omega(n)-1} (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 A381641 #8 Mar 03 2025 13:34:02
%S A381641 0,0,0,0,0,1,0,0,0,3,0,1,0,5,2,0,0,1,0,3,4,9,0,1,0,11,0,5,0,11,0,0,8,
%T A381641 15,2,1,0,17,10,3,0,19,0,9,2,21,0,1,0,3,14,11,0,1,6,5,16,27,0,11,0,29,
%U A381641 4,0,8,35,0,15,20,31,0,1,0,35,2,17,4,43,0,3,0
%N A381641 Numerators of Sum_{i=1..omega(n)-1} (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 A381641 Amiram Eldar, <a href="/A381641/b381641.txt">Table of n, a(n) for n = 1..10000</a>
%H A381641 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 A381641 a(n) = 0 if and only if n is a power of a prime (A000961).
%F A381641 a(n)/A381639(n) = A001221(n) - 1 - A381638(n)/A381639(n).
%e A381641 Fractions begin with 0, 0, 0, 0, 0, 1/3, 0, 0, 0, 3/5, 0, 1/3, ...
%t A381641 a[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, Numerator[Total[1 - Most[p]/Rest[p]]]]; Array[a, 100]
%o A381641 (PARI) a(n) = {my(p = factor(n)[,1]); numerator(sum(i = 1, #p-1, 1 - p[i]/p[i+1]));}
%Y A381641 Cf. A000961, A001221, A381638, A381639 (denominators).
%K A381641 nonn,easy,frac
%O A381641 1,10
%A A381641 _Amiram Eldar_, Mar 03 2025