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 A069205 #57 Aug 20 2024 01:16:02
%S A069205 1,3,5,9,11,15,17,25,29,33,35,43,45,49,53,69,71,79,81,89,93,97,99,115,
%T A069205 119,123,131,139,141,149,151,183,187,191,195,211,213,217,221,237,239,
%U A069205 247,249,257,265,269,271,303,307,315,319,327,329,345,349,365,369,373
%N A069205 a(n) = Sum_{k=1..n} 2^bigomega(k).
%C A069205 Partial sums of A061142. - _Michel Marcus_, Aug 08 2017
%D A069205 G. Tenenbaum and Jie Wu, Cours Spécialisés No. 2: "Théorie analytique et probabiliste des nombres", Collection SMF, Ordres moyens, p. 20.
%D A069205 G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 53, exercise 5 (in the third edition 2015, p. 59, exercise 57).
%H A069205 Michel Marcus, <a href="/A069205/b069205.txt">Table of n, a(n) for n = 1..1000</a>
%H A069205 Emil Grosswald, <a href="http://doi.org/10.1215/S0012-7094-56-02305-5">The average order of an arithmetic function</a>, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44.
%H A069205 Vaclav Kotesovec, <a href="/A069205/a069205_2.jpg">Graph - the asymptotic ratio (10^8 terms)</a>
%F A069205 Asymptotic formula: a(n) = 1/(8*log(2))*C*n*log(n)^2+O(n*log(n)) with C = A167864 = Product_{p primes > 2} (1+1/p/(p-2)) where the product is over all the primes p>2.
%F A069205 From _Daniel Suteu_, May 23 2020: (Start)
%F A069205 a(n) = Sum_{k=1..n} 2^(bigomega(k) - omega(k)) * floor(n/k).
%F A069205 a(n) = Sum_{k=1..n} A335073(floor(n/k)).
%F A069205 a(n) = 1 + Sum_{k=1..floor(log_2(n))} 2^k * pi_k(n), where pi_k(n) is the number of k-almost primes <= n. (End)
%F A069205 More precise asymptotics [Grosswald, 1956]: a(n) ~ A167864*n*log(n)*(log(n) - 2 - 4*A347195 + 4*gamma + 5*log(2))/(8*log(2)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Aug 22 2021
%F A069205 Even more precise formula: a(n) ~ A167864 * n / (8*log(2)) * (log(n)^2 + (4*g + 5*log(2) - 2 - 4*A347195)*log(n) + 2 + 2*g^2 - 4*sg1 - 5*log(2) + 13*log(2)^2/6 + 2*g*(5*log(2) - 2) - 2*A347195*(5*log(2) - 2 + 4*g) + 4*A347195^2 + c), where c = Sum_{prime p > 2} (2*p * (2*p-3)* log(p)^2) / ((p-2)^2 * (p-1)^2) = 8.86809160013722347937514407919207620377461987744681170588044228288988578547..., g is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - _Vaclav Kotesovec_, Feb 11 2022
%t A069205 Accumulate[2^PrimeOmega[Range[60]]] (* _Harvey P. Dale_, Aug 22 2011 *)
%o A069205 (PARI) a(n) = sum(k=1, n, 2^bigomega(k)); \\ _Michel Marcus_, Aug 08 2017
%Y A069205 Cf. A061142, A167864, A347195, A350961.
%K A069205 easy,nonn
%O A069205 1,2
%A A069205 _Benoit Cloitre_, Apr 14 2002