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.

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

Offset: 1

Author

Benoit Cloitre, Apr 14 2002

Keywords

• easy
• nonn

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

Crossrefs

Partial sums of A074823.

Cf. A008966, A034444, A065473, A347149.

Programs

Magma

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

PARI

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

Scheme

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

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

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*gamma - 1) + f'(1)), where f(1) = A065473, f'(1) = f(1) * Sum_{primes p} 6*log(p)/(p^2 + p - 2) = 0.802323384763097462846799913287578352653695442033314074501634920897596526... and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 20 2021