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A069212
a(n) = Sum_{k=1..n} 3^omega(k).
6
1, 4, 7, 10, 13, 22, 25, 28, 31, 40, 43, 52, 55, 64, 73, 76, 79, 88, 91, 100, 109, 118, 121, 130, 133, 142, 145, 154, 157, 184, 187, 190, 199, 208, 217, 226, 229, 238, 247, 256, 259, 286, 289, 298, 307, 316, 319, 328, 331, 340, 349, 358, 361, 370, 379, 388, 397
OFFSET
1,2
COMMENTS
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)).
REFERENCES
G. Tenenbaum and Jie Wu, Cours Spécialisés No. 2: "Théorie analytique et probabiliste des nombres", Collection SMF, Ordres moyens, p. 20.
G. Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Soc. (2015). See page 59.
FORMULA
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.
The constant C is A065473/2. - Amiram Eldar, May 24 2020
From Ridouane Oudra, Jan 01 2021: (Start)
a(n) = Sum_{i=1..n} Sum_{j=1..n} mu(i*j)^2*floor(n/(i*j));
a(n) = Sum_{i=1..n} mu(i)^2*tau(i)*floor(n/i);
a(n) = Sum_{i=1..n} 2^Omega(i)*mu(i)^2*floor(n/i), where Omega = A001222. (End)
From Vaclav Kotesovec, Feb 16 2022: (Start)
More precise asymptotics:
Let f(s) = Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)), then
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),
where f(1) = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.2867474284344787341078927127898384464343318440970569956414778593366522...,
f'(1) = f(1) * Sum_{primes p} 6*log(p) / (p^2 + p - 2) = 0.8023233847630974628467999132875783526536954420333140745016349208975965...,
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...
and gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)
MATHEMATICA
Accumulate @ Table[3^PrimeNu[n], {n, 1, 57}] (* Amiram Eldar, May 24 2020 *)
PROG
(Python)
from sympy.ntheory.factor_ import primenu
def A069212(n): return sum(3**primenu(m) for m in range(1, n+1)) # Chai Wah Wu, Sep 07 2023
CROSSREFS
Partial sums of A074816.
Sequence in context: A008470 A002640 A096675 * A091290 A119256 A143454
KEYWORD
easy,nonn
AUTHOR
Benoit Cloitre, Apr 14 2002
STATUS
approved