cp's OEIS Frontend

A069212 a(n) = Sum_{k=1..n} 3^omega(k).

%I A069212 #40 Sep 07 2023 12:50:10
%S A069212 1,4,7,10,13,22,25,28,31,40,43,52,55,64,73,76,79,88,91,100,109,118,
%T A069212 121,130,133,142,145,154,157,184,187,190,199,208,217,226,229,238,247,
%U A069212 256,259,286,289,298,307,316,319,328,331,340,349,358,361,370,379,388,397
%N A069212 a(n) = Sum_{k=1..n} 3^omega(k).
%C A069212 More generally, if b is an integer =>3, Sum_{k=1..n} b^omega(k) ~ C(b)*n*log(n)^(b-1) where C(b)=1/(b-1)!*prod((1-1/p)^(b-1)*(1+(b-1)/p)).
%D A069212 G. Tenenbaum and Jie Wu, Cours Spécialisés No. 2: "Théorie analytique et probabiliste des nombres", Collection SMF, Ordres moyens, p. 20.
%D A069212 G. Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Soc. (2015). See page 59.
%H A069212 Amiram Eldar, <a href="/A069212/b069212.txt">Table of n, a(n) for n = 1..10000</a>
%H A069212 Vaclav Kotesovec, <a href="/A069212/a069212.jpg">Graph - the asymptotic ratio (1000000 terms)</a>
%F A069212 Asymptotic formula: a(n) ~ C*n*log(n)^2 with C = (1/2) * Product_{p} ((1-1/p)^2*(1+2/p)) where the product is over all the primes.
%F A069212 The constant C is A065473/2. - _Amiram Eldar_, May 24 2020
%F A069212 From _Ridouane Oudra_, Jan 01 2021: (Start)
%F A069212 a(n) = Sum_{i=1..n} Sum_{j=1..n} mu(i*j)^2*floor(n/(i*j));
%F A069212 a(n) = Sum_{i=1..n} mu(i)^2*tau(i)*floor(n/i);
%F A069212 a(n) = Sum_{i=1..n} 2^Omega(i)*mu(i)^2*floor(n/i), where Omega = A001222. (End)
%F A069212 From _Vaclav Kotesovec_, Feb 16 2022: (Start)
%F A069212 More precise asymptotics:
%F A069212 Let f(s) = Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)), then
%F A069212 a(n) ~ n * (f(1)*log(n)^2/2 + log(n)*((3*gamma - 1)*f(1) + f'(1)) + f(1)*(1 - 3*gamma + 3*gamma^2 - 3*sg1) + (3*gamma - 1)*f'(1) + f''(1)/2),
%F A069212 where f(1) = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.2867474284344787341078927127898384464343318440970569956414778593366522...,
%F A069212 f'(1) = f(1) * Sum_{primes p} 6*log(p) / (p^2 + p - 2) = 0.8023233847630974628467999132875783526536954420333140745016349208975965...,
%F A069212 f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} -6*p*(2*p+1) * log(p)^2 / (p^2 + p - 2)^2 = -0.255987592484328884627082229528266165335336670389046663124468278519...
%F A069212 and gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)
%t A069212 Accumulate @ Table[3^PrimeNu[n], {n, 1, 57}] (* _Amiram Eldar_, May 24 2020 *)
%o A069212 (Python)
%o A069212 from sympy.ntheory.factor_ import primenu
%o A069212 def A069212(n): return sum(3**primenu(m) for m in range(1,n+1)) # _Chai Wah Wu_, Sep 07 2023
%Y A069212 Partial sums of A074816.
%Y A069212 Cf. A001222, A064608, A065473, A350961.
%K A069212 easy,nonn
%O A069212 1,2
%A A069212 _Benoit Cloitre_, Apr 14 2002