This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A346009 #14 Jul 14 2025 15:15:13
%S A346009 0,1,1,2,1,1,1,3,2,1,1,7,1,1,1,4,1,7,1,7,1,1,1,5,2,1,3,7,1,3,1,5,1,1,
%T A346009 1,4,1,1,1,5,1,3,1,7,7,1,1,13,2,7,1,7,1,5,1,5,1,1,1,5,1,1,7,6,1,3,1,7,
%U A346009 1,3,1,17,1,1,7,7,1,3,1,13,4,1,1,5,1,1
%N A346009 a(n) is the numerator of the average number of distinct prime factors of the divisors of n.
%D A346009 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.3.21 on page 100.
%H A346009 Amiram Eldar, <a href="/A346009/b346009.txt">Table of n, a(n) for n = 1..10000</a>
%H A346009 R. L. Duncan, <a href="https://www.jstor.org/stable/2311587">Note on the divisors of a number</a>, The American Mathematical Monthly, Vol. 68, No. 4 (1961), pp. 356-359.
%H A346009 Sébastien Gaboury, <a href="http://hdl.handle.net/20.500.11794/18899">Sur les convolutions de fonctions arithmétiques</a>, M.Sc. thesis, Laval University, Quebec, 2007.
%F A346009 Let f(n) = a(n)/A346010(n) be the sequence of fractions. Then:
%F A346009 f(n) = A062799(n)/A000005(n).
%F A346009 f(n) = (Sum_{p prime, p|n} d(n/p))/d(n), where d(n) is the number of divisors of n (A000005).
%F A346009 f(n) depends only on the prime signature of n: If n = Product_{i} p_i^e_i, then a(n) = Sum_{i} e_i/(e_i + 1).
%F A346009 f(p) = 1/2 for prime p.
%F A346009 f(n) = 1 for squarefree semiprimes n (A006881).
%F A346009 Sum_{k=1..n} f(k) ~ (1/2) * A013939(n) + C*n + O(n/log(n)) ~ n*log(log(n))/2 + (B/2 + C)*n + O(n/log(n)), where B is Mertens's constant (A077761) and C = A346011 (Duncan, 1961).
%e A346009 The fractions begin with 0, 1/2, 1/2, 2/3, 1/2, 1, 1/2, 3/4, 2/3, 1, 1/2, 7/6, ...
%e A346009 f(2) = 1/2 since 2 has 2 divisors, 1 and 2, and (omega(1) + omega(2))/2 = (0 + 1)/2 = 1/2.
%e A346009 f(6) = 1 since 6 has 4 divisors, 1, 2, 3 and 6 and (omega(1) + omega(2) + omega(3) + omega(6))/4 = (0 + 1 + 1 + 2)/4 = 1.
%t A346009 a[n_] := Numerator[DivisorSum[n, PrimeNu[#] &]/DivisorSigma[0, n]]; Array[a, 100]
%t A346009 (* or *)
%t A346009 f[p_, e_] := e/(e+1); a[1] = 0; a[n_] := Numerator[Plus @@ f @@@ FactorInteger[n]]; Array[a, 100]
%Y A346009 Cf. A000005, A001221, A006881, A013939, A062799, A077761, A346010 (denominators), A346011.
%K A346009 nonn,frac
%O A346009 1,4
%A A346009 _Amiram Eldar_, Jul 01 2021