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

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

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.

Cf. A001222, A064608, A065473, A350961.

Sequence in context: A008470 A002640 A096675 * A091290 A119256 A143454

Adjacent sequences: A069209 A069210 A069211 * A069213 A069214 A069215

Keyword

easy,nonn

Author

Benoit Cloitre, Apr 14 2002

Status

approved