A069205 a(n) = Sum_{k=1..n} 2^bigomega(k).

1, 3, 5, 9, 11, 15, 17, 25, 29, 33, 35, 43, 45, 49, 53, 69, 71, 79, 81, 89, 93, 97, 99, 115, 119, 123, 131, 139, 141, 149, 151, 183, 187, 191, 195, 211, 213, 217, 221, 237, 239, 247, 249, 257, 265, 269, 271, 303, 307, 315, 319, 327, 329, 345, 349, 365, 369, 373

Offset: 1

Author

Benoit Cloitre, Apr 14 2002

Keywords

• easy
• nonn

Comments

Partial sums of A061142. - Michel Marcus, Aug 08 2017

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, Cambridge University Press, 1995, p. 53, exercise 5 (in the third edition 2015, p. 59, exercise 57).

Links

• Michel Marcus, Table of n, a(n) for n = 1..1000
• Emil Grosswald, The average order of an arithmetic function, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44.
• Vaclav Kotesovec, Graph - the asymptotic ratio (10^8 terms)

Crossrefs

Cf. A061142, A167864, A347195, A350961.

Programs

Mathematica

Accumulate[2^PrimeOmega[Range[60]]] (* Harvey P. Dale, Aug 22 2011 *)

PARI

a(n) = sum(k=1, n, 2^bigomega(k)); \\ Michel Marcus, Aug 08 2017

Formula

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.

From Daniel Suteu, May 23 2020: (Start)

a(n) = Sum_{k=1..n} 2^(bigomega(k) - omega(k)) * floor(n/k).

a(n) = Sum_{k=1..n} A335073(floor(n/k)).

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)

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

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