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A028235 If n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j (the denominator of this sum is A007947).

%I A028235 #33 Jul 08 2025 18:30:57
%S A028235 0,1,1,1,1,5,1,1,1,7,1,5,1,9,8,1,1,5,1,7,10,13,1,5,1,15,1,9,1,31,1,1,
%T A028235 14,19,12,5,1,21,16,7,1,41,1,13,8,25,1,5,1,7,20,15,1,5,16,9,22,31,1,
%U A028235 31,1,33,10,1,18,61,1,19,26,59,1,5,1,39,8,21,18,71,1,7,1,43,1,41,22,45,32
%N A028235 If n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j (the denominator of this sum is A007947).
%C A028235 For n=1, the empty sum = 0 = 0/1 = a(1)/A007947(1), thus a(1) should be 0. - _Antti Karttunen_, Mar 04 2018
%H A028235 Antti Karttunen, <a href="/A028235/b028235.txt">Table of n, a(n) for n = 1..65537</a>
%F A028235 Fraction is additive with a(p^e) = 1/p.
%F A028235 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A007947(k) = Sum_{p prime} 1/p^2 = 0.452247... (A085548). - _Amiram Eldar_, Sep 29 2023
%F A028235 a(n) = A003415(A007947(n)) = A069359(A007947(n)). - _Antti Karttunen_, Jan 22 2025
%e A028235 Fractions begin with 0, 1/2, 1/3, 1/2, 1/5, 5/6, 1/7, 1/2, 1/3, 7/10, 1/11, 5/6, ...
%t A028235 a[1] = 0; a[n_] := 1/FactorInteger[n][[All, 1]] // Total // Numerator;
%t A028235 Array[a, 100] (* _Jean-François Alcover_, May 08 2019 *)
%o A028235 (PARI) A028235(n) = numerator(vecsum(apply(p->(1/p), factor(n)[, 1]))); \\ _Antti Karttunen_, Mar 04 2018
%Y A028235 Cf. A007947 (denominators), A003415, A069359, A085548, A379967.
%K A028235 nonn,frac,easy
%O A028235 1,6
%A A028235 _N. J. A. Sloane_
%E A028235 More terms from _Erich Friedman_.
%E A028235 Term a(1) changed to 0 by _Antti Karttunen_, Mar 04 2018