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A069212 a(n) = Sum_{k=1..n} 3^omega(k). 2
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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

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

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

MATHEMATICA

Accumulate @ Table[3^PrimeNu[n], {n, 1, 57}] (* Amiram Eldar, May 24 2020 *)

CROSSREFS

Partial sums of A074816.

Cf. A001222, A065473.

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

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Last modified August 7 22:44 EDT 2020. Contains 336279 sequences. (Running on oeis4.)