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A069201 a(n) = Sum_{k=1..n} mu(k)^2 * 2^omega(k) where omega(k) is the number of distinct primes in the factorization of k. 3
1, 3, 5, 5, 7, 11, 13, 13, 13, 17, 19, 19, 21, 25, 29, 29, 31, 31, 33, 33, 37, 41, 43, 43, 43, 47, 47, 47, 49, 57, 59, 59, 63, 67, 71, 71, 73, 77, 81, 81, 83, 91, 93, 93, 93, 97, 99, 99, 99, 99, 103, 103, 105, 105, 109, 109, 113, 117, 119, 119, 121, 125, 125, 125, 129, 137 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

G. Tenenbaum and Jie Wu, Cours Spécialisés No. 2: "Théorie analytique et probabiliste des nombres", Collection SMF, Ordres moyens, p. 20.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

FORMULA

Asymptotic formula: a(n) = C*n*log(n) + O(n) with C = Product_{p prime} (1 - 1/p)^2*(1 + 2/p).

The constant C is A065473. - Amiram Eldar, May 24 2020

a(n) = Sum_{k=1..n} mu(k)^2*d(k), where d is the number of divisors function (A000005). - Ridouane Oudra, Jul 25 2019

MAPLE

with(numtheory): seq(add(tau(k)*mobius(k)^2, k=1..n), n=1..90); # Ridouane Oudra, Jul 25 2019

MATHEMATICA

Accumulate @ Table[MoebiusMu[n]^2 * 2^PrimeNu[n], {n, 1, 66}] (* Amiram Eldar, May 24 2020 *)

PROG

(Scheme) (define (A069201 n) (if (= 1 n) n (+ (A074823 n) (A069201 (- n 1))))) ;; Antti Karttunen, Jul 23 2017

(PARI) a(n) = sum(k=1, n, moebius(k)^2*2^omega(k)); \\ Michel Marcus, Jul 23 2017

(MAGMA) [&+[MoebiusMu(k)^2*#Divisors(k):k in [1..n]]: n in [1..66]]; // Marius A. Burtea, Jul 27 2019

CROSSREFS

Partial sums of A074823.

Cf. A008966, A034444, A065473.

Sequence in context: A227950 A063202 A058020 * A272882 A077800 A073340

Adjacent sequences:  A069198 A069199 A069200 * A069202 A069203 A069204

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, Apr 14 2002

STATUS

approved

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Last modified September 30 14:46 EDT 2020. Contains 337439 sequences. (Running on oeis4.)