A347149 -id:A347149 - cp's OEIS frontend

A165824 Totally multiplicative sequence with a(p) = 3.

1, 3, 3, 9, 3, 9, 3, 27, 9, 9, 3, 27, 3, 9, 9, 81, 3, 27, 3, 27, 9, 9, 3, 81, 9, 9, 27, 27, 3, 27, 3, 243, 9, 9, 9, 81, 3, 9, 9, 81, 3, 27, 3, 27, 27, 9, 3, 243, 9, 27, 9, 27, 3, 81, 9, 81, 9, 9, 3, 81, 3, 9, 27, 729, 9, 27, 3, 27, 9, 27, 3, 243, 3, 9, 27, 27

Offset: 1

Jaroslav Krizek, Sep 28 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000244, A001222, A061142, A347149, A350961 (partial sums).

A165824 := proc(n)
    3^numtheory[bigomega](n) ;
end proc:
seq(A165824(n),n=1..40) ; # R. J. Mathar, Mar 07 2022

3^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 09 2016 *)

a(n) = 3^bigomega(n); \\ Altug Alkan, Apr 09 2016

for(n=1, 100, print1(direuler(p=2, n, 1/(1-3*X))[n], ", ")) \\ Vaclav Kotesovec, May 08 2025

a(n) = A000244(A001222(n)) = 3^bigomega(n) = 3^A001222(n).

Dirichlet g.f.: Product_{p prime} 1 / (1 - 3 * p^(-s)). - Ilya Gutkovskiy, Oct 30 2019

More terms from Vaclav Kotesovec, Feb 16 2022

A074823 a(n) = 2^omega(n)*mu(n)^2.

Original entry on oeis.org

1, 2, 2, 0, 2, 4, 2, 0, 0, 4, 2, 0, 2, 4, 4, 0, 2, 0, 2, 0, 4, 4, 2, 0, 0, 4, 0, 0, 2, 8, 2, 0, 4, 4, 4, 0, 2, 4, 4, 0, 2, 8, 2, 0, 0, 4, 2, 0, 0, 0, 4, 0, 2, 0, 4, 0, 4, 4, 2, 0, 2, 4, 0, 0, 4, 8, 2, 0, 4, 8, 2, 0, 2, 4, 0, 0, 4, 8, 2, 0, 0, 4, 2, 0, 4, 4, 4, 0, 2, 0, 4, 0, 4, 4, 4, 0, 2, 0, 0, 0, 2, 8, 2, 0, 8

Offset: 1

Benoit Cloitre, Sep 08 2002

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A008683, A008966, A034444, A069201, A226177, A347149.

Table[MoebiusMu[n]^2 * 2^PrimeNu[n], {n, 1, 100}] (* Vaclav Kotesovec, Aug 20 2021 *)
f[p_, e_] :=If[e==1, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)

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

for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X))[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

for(n=1, 100, print1(direuler(p=2, n, (1 - 3*X^2 + 2*X^3)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

(define (A074823 n) (if (= 1 n) n (* (if (= 1 (A067029 n)) 2 0) (A074823 (A028234 n))))) ;; Antti Karttunen, Jul 23 2017

Sum_{k=1..n} a(k) = A069201(n).
Multiplicative with a(p)=2, a(p^e)=0, e > 1.
a(n) = A034444(n)*A008966(n). - R. J. Mathar, Apr 15 2011
Sum_{n>0} a(n)/n^s = Product_{p prime} (1 + 2p^(-s)). - Ralf Stephan, Jul 07 2013
a(n) = abs(A226177(n)). - Antti Karttunen, Jul 23 2017
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - Vaclav Kotesovec, Aug 20 2021

Additional comments from Vladeta Jovovic, Dec 30 2002

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.

Original entry on oeis.org

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

Benoit Cloitre, Apr 14 2002

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

Antti Karttunen, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Partial sums of A074823.

Cf. A008966, A034444, A065473, A347149.

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

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

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

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

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

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

A351521 Dirichlet g.f.: Product_{p prime} (1 + 4*p^(-s)).

Original entry on oeis.org

1, 4, 4, 0, 4, 16, 4, 0, 0, 16, 4, 0, 4, 16, 16, 0, 4, 0, 4, 0, 16, 16, 4, 0, 0, 16, 0, 0, 4, 64, 4, 0, 16, 16, 16, 0, 4, 16, 16, 0, 4, 64, 4, 0, 0, 16, 4, 0, 0, 0, 16, 0, 4, 0, 16, 0, 16, 16, 4, 0, 4, 16, 0, 0, 16, 64, 4, 0, 16, 64, 4, 0, 4, 16, 0, 0, 16, 64

Offset: 1

Vaclav Kotesovec, Feb 17 2022

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A074823, A347149.
Cf. A061142, A165824, A165825.
Cf. A008966, A035116.

Table[MoebiusMu[n]^2 * 4^PrimeNu[n], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X))[n], ", "))

Dirichlet g.f.: zeta(s)^4 * Product_{prime p} (1 + (4 - 15*p^s + 20*p^(2*s) - 10*p^(3*s))/p^(5*s)).
a(n) = A008966(n) * A035116(n). - Enrique Pérez Herrero, Oct 27 2022
Multiplicative with a(p) = 4, and a(p^e) = 0 for e >= 2. - Amiram Eldar, Dec 25 2022

A349924 Dirichlet g.f.: Product_{k>=2} (1 + 3 * k^(-s)).

Original entry on oeis.org

1, 3, 3, 3, 3, 12, 3, 12, 3, 12, 3, 21, 3, 12, 12, 12, 3, 21, 3, 21, 12, 12, 3, 57, 3, 12, 12, 21, 3, 57, 3, 21, 12, 12, 12, 57, 3, 12, 12, 57, 3, 57, 3, 21, 21, 12, 3, 93, 3, 21, 12, 21, 3, 57, 12, 57, 12, 12, 3, 129, 3, 12, 21, 48, 12, 57, 3, 21, 12, 57

Offset: 1

Ilya Gutkovskiy, Dec 05 2021

nonn

Cf. A032308, A045778, A339335, A347149, A349923.

cp's OEIS Frontend

A165824 Totally multiplicative sequence with a(p) = 3.

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A074823 a(n) = 2^omega(n)*mu(n)^2.

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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.

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A351521 Dirichlet g.f.: Product_{p prime} (1 + 4*p^(-s)).

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A349924 Dirichlet g.f.: Product_{k>=2} (1 + 3 * k^(-s)).

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