A074823 -id:A074823 - cp's OEIS frontend

A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).

1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Oct 14 2002

Multiplicative: a(1) = 1, a(n) for n >=2 is sign of parity of number of distinct primes dividing n. a(p) = -1, a(pq) = 1, a(pq...z) = (-1)^k, a(p^k) = -1, where p,q,.. z are distinct primes and k natural numbers. - Jaroslav Krizek, Mar 17 2009

a(n) is the unitary Moebius function, i.e., the inverse of the constant 1 function under the unitary convolution defined by (f X g)(n)= sum of f(d)g(n/d), where the sum is over the unitary divisors d of n (divisors d of n such that gcd(d,n/d)=1). - Laszlo Toth, Oct 08 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr., Vol. 74 (1960), pp. 66-80.
Jan van de Lune and Robert E. Dressler, Some theorems concerning the number theoretic function omega(n), Journal für die reine und angewandte Mathematik, Vol. 277 (1975), pp. 117-119; alternative link.

Cf. A000005, A000010, A001221, A007947, A008683, A008836, A030230, A065469, A076480, A180403, A226177.

Cf. A174863 (partial sums).

a076479 = a008683 . a007947  -- Reinhard Zumkeller, Jun 01 2013

[(-1)^(#PrimeDivisors(n)): n in [1..100]]; // Vincenzo Librandi, Dec 31 2018

A076479 := proc(n)
    (-1)^A001221(n) ;
end proc:
seq(A076479(n),n=1..80) ; # R. J. Mathar, Nov 02 2016

Table[(-1)^PrimeNu[n], {n,50}] (* Enrique Pérez Herrero, Jan 17 2013 *)

N=66;
mu=vector(N); mu[1]=1;
{ for (n=2,N,
    s = 0;
    fordiv (n,d,
        if (gcd(d,n/d)!=1, next() ); /* unitary divisors only */
        s += mu[d];
    );
    mu[n] = -s;
); };
mu /* Joerg Arndt, May 13 2011 */
/* omitting the line if ( gcd(...)) gives the usual Moebius function */

a(n)=(-1)^omega(n) \\ Charles R Greathouse IV, Aug 02 2013

from math import prod
from sympy.ntheory import mobius, primefactors
def A076479(n): return mobius(prod(primefactors(n))) # Chai Wah Wu, Sep 24 2021

a(n) = A008683(A007947(n)).
a(n) = (-1)^A001221(n). Multiplicative with a(p^e) = -1. - Vladeta Jovovic, Dec 03 2002
a(n) = sign(A180403(n)). - Mats Granvik, Oct 08 2010
Sum_{n>=1} a(n)*phi(n)/n^3 = A065463 with phi()=A000010() [Cohen, Lemma 3.5]. - R. J. Mathar, Apr 11 2011
Dirichlet convolution of A000012 with A226177 (signed variant of A074823 with one factor mu(n) removed). - R. J. Mathar, Apr 19 2011
Sum_{n>=1} a(n)/n^2 = A065469. - R. J. Mathar, Apr 19 2011
a(n) = Sum_{d|n} mu(d)*tau_2(d) = Sum_{d|n} A008683(d)*A000005(d) . - Enrique Pérez Herrero, Jan 17 2013
a(A030230(n)) = -1; a(A030231(n)) = +1. - Reinhard Zumkeller, Jun 01 2013
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 2p^(-s)). - Álvar Ibeas, Dec 30 2018
Sum_{n>=1} a(n)/n = 0 (van de Lune and Dressler, 1975). - Amiram Eldar, Mar 05 2021
From Richard L. Ollerton, May 07 2021: (Start)
For n>1, Sum_{k=1..n} a(gcd(n,k))*phi(gcd(n,k))*rad(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*phi(gcd(n,k))*rad(n/gcd(n,k))*gcd(n,k) = 0. (End)
a(n) = Sum_{d1|n} Sum_{d2|n} mu(d1*d2). - Ridouane Oudra, May 25 2023

A065473 Decimal expansion of the strongly carefree constant: Product_{p prime} (1 - (3*p-2)/(p^3)).

Original entry on oeis.org

2, 8, 6, 7, 4, 7, 4, 2, 8, 4, 3, 4, 4, 7, 8, 7, 3, 4, 1, 0, 7, 8, 9, 2, 7, 1, 2, 7, 8, 9, 8, 3, 8, 4, 4, 6, 4, 3, 4, 3, 3, 1, 8, 4, 4, 0, 9, 7, 0, 5, 6, 9, 9, 5, 6, 4, 1, 4, 7, 7, 8, 5, 9, 3, 3, 6, 6, 5, 2, 2, 4, 3, 1, 3, 1, 9, 4, 3, 2, 5, 8, 2, 4, 8, 9, 1, 2, 6, 8, 2, 5, 5, 3, 7, 4, 2, 3, 7, 4, 6, 8, 5, 3, 6, 4, 7

Offset: 0

N. J. A. Sloane, Nov 19 2001

Also decimal expansion of the probability that an integer triple (x, y, z) is pairwise coprime. - Charles R Greathouse IV, Nov 14 2011

The probability that 2 numbers chosen at random are coprime, and both squarefree (Delange, 1969). - Amiram Eldar, Aug 04 2020

			0.2867474284344787341078927127898384...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6, p. 41.
Gerald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd edition, American Mathematical Society, 2015, page 59, exercise 55 and 56.

Juan Arias de Reyna, R. Heyman, Counting Tuples Restricted by Pairwise Coprimality Conditions, J. Int. Seq. 18 (2015) 15.10.4
Tim Browning, The divisor problem for binary cubic forms, Journal de théorie des nombres de Bordeaux, Vol. 23, No. 3 (2011), pp. 579-602; arXiv preprint, arXiv:1006.3476 [math.NT], 2010.
Hubert Delange, On some sets of pairs of positive integers, Journal of Number Theory, Vol. 1, No. 3 (1969), pp. 261-279. See p. 277.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 181.
G. Niklasch, Some number theoretical constants: 1000-digit values.
G. Niklasch, Some number theoretical constants: 1000-digit values. [cached copy]
László Tóth, The probability that k positive integers are pairwise relatively prime, Fibonacci Quart., Vol. 40 (2002), pp. 13-18.
László Tóth, Another generalization of Euler's arithmetic function and of Menon's identity, arXiv:2006.12438 [math.NT], 2020. See p. 3.
Eric Weisstein's World of Mathematics, Carefree Couple.

Cf. A065472, A069201, A069212, A074816, A074823, A078073, A098198, A227929, A299822.

digits = 100; NSum[-(2+(-2)^n)*PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2 digits, WorkingPrecision -> 2 digits, Method -> "AlternatingSigns"] // Exp // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 11 2016 *)

prodeulerrat(1 - (3*p-2)/(p^3)) \\ Amiram Eldar, Mar 17 2021

Equals Prod_{p prime} (1 - 1/p)^2*(1 + 2/p). - Michel Marcus, Apr 16 2016
The constant c in Sum_{k<=x} mu(k)^2 * 2^omega(k) = c * x * log(x) + O(x), where mu is A008683 and omega is A001221, and in Sum_{k<=x} 3^omega(k) = (1/2) * c * x * log(x)^2 + O(x*log(x)) (see Tenenbaum, 2015). - Amiram Eldar, May 24 2020
Equals A065472 * A227929 =  A065472 / A098198. - Amiram Eldar, Aug 04 2020

Name corrected by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 03 2003

More digits from Vaclav Kotesovec, Dec 19 2019

A074816 a(n) = 3^A001221(n) = 3^omega(n).

Original entry on oeis.org

1, 3, 3, 3, 3, 9, 3, 3, 3, 9, 3, 9, 3, 9, 9, 3, 3, 9, 3, 9, 9, 9, 3, 9, 3, 9, 3, 9, 3, 27, 3, 3, 9, 9, 9, 9, 3, 9, 9, 9, 3, 27, 3, 9, 9, 9, 3, 9, 3, 9, 9, 9, 3, 9, 9, 9, 9, 9, 3, 27, 3, 9, 9, 3, 9, 27, 3, 9, 9, 27, 3, 9, 3, 9, 9, 9, 9, 27, 3, 9, 3, 9, 3, 27, 9, 9, 9, 9, 3, 27, 9, 9, 9, 9, 9, 9, 3, 9, 9, 9

Offset: 1

Benoit Cloitre, Sep 08 2002

Old name was: a(n) = sum(d|n, tau(d)*mu(d)^2 ).
Terms are powers of 3.
The inverse Mobius transform of A074823, as the Dirichlet g.f. is product_{primes p} (1+2*p^(-s)) and the Dirichlet g.f. of A074816 is product_{primes p} (1+2*p^(-s))/(1-p^(-s)). - R. J. Mathar, Feb 09 2011
If n is squarefree, then a(n) = #{(x, y) : x, y positive integers, lcm (x, y) = n}. See Crandall & Pomerance. - Michel Marcus, Mar 23 2016

Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 2.3 p. 108.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001221, A034444, A069212, A124508.

A074816[n_]:=3^PrimeNu[n]; (* Enrique Pérez Herrero, Jun 28 2010 *)

a(n) = 3^omega(n); \\ Michel Marcus, Mar 23 2016

for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Feb 16 2022

a(n) = 3^m if n is divisible by m distinct primes. i.e., a(n)=3 if n is in A000961; a(n)=9 if n is in A007774 ...
a(n) = 3^A001221(n) = 3^omega(n). Multiplicative with a(p^e)=3. - Vladeta Jovovic, Sep 09 2002.
a(n) = tau_3(rad(n)) = A007425(A007947(n)). - Enrique Pérez Herrero, Jun 24 2010
a(n) = abs(Sum_{d|n} A000005(d^3)*mu(d)). - Enrique Pérez Herrero, Jun 28 2010
a(n) = Sum_{d|n, gcd(d, n/d) = 1} 2^omega(d) (The total number of unitary divisors of the unitary divisors of n). - Amiram Eldar, May 29 2020, Dec 27 2024
a(n) = Sum_{d1|n, d2|n} mu(d1*d2)^2. - Wesley Ivan Hurt, Feb 04 2022
Dirichlet g.f.: zeta(s)^3 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - Vaclav Kotesovec, Feb 16 2022

Simpler definition at the suggestion of Michel Marcus. - N. J. A. Sloane, Mar 25 2016

A226177 a(n) = mu(n)*d(n), where mu(n) = A008683 and d(n) = A000005.

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

Wesley Ivan Hurt, May 29 2013

The prime numbers are the only solutions to mu(n)*d(n) = -2.
Multiplicative with a(p) = -2, a(p^e) = 0, e > 1.
The Moebius inverse is A076479, and the Dirichlet inverse A061142. - R. J. Mathar, Jun 03 2013
Möbius transform of (-1)^omega(n). - Wesley Ivan Hurt, Jun 22 2024

			a(5) = mu(5)*d(5) = (-1)(2) = -2.

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

Cf. A000005, A000040, A001358, A008683, A074823 (absolute values), A001221.

with(numtheory); a:=n->mobius(n)*tau(n); seq(a(k),k=1..100);

Table[MoebiusMu[n] DivisorSigma[0, n], {n, 105}] (* Michael De Vlieger, Jul 23 2017 *)

A226177(n) = moebius(n)*numdiv(n); \\ Antti Karttunen, Jul 23 2017

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X))[n], ", ")); \\ Vaclav Kotesovec, Aug 21 2021

(define (A226177 n) (if (= 1 n) n (* (if (= 1 (A067029 n)) -2 0) (A226177 (A028234 n))))) ;; Antti Karttunen, Jul 23 2017

a(n) = mu(n)*d(n) = A008683(n)*A000005(n).
Sum_{n>0} a(n)/n^s = Product_{p prime} (1 - 2p^(-s)). - Ralf Stephan, Jul 07 2013
a(n) = mu(n) * 2^omega(n) = |mu(n)| * (-2)^omega(n), where omega = A001221. - Álvar Ibeas, Dec 30 2018
a(n) = Sum_{d|n} (-1)^omega(d) * mu(n/d). - Wesley Ivan Hurt, Jun 22 2024

More terms from Antti Karttunen, Jul 23 2017

Name changed by David A. Corneth, Jul 23 2017

A347149 Dirichlet g.f.: Product_{primes p} (1 + 3/p^s).

Original entry on oeis.org

1, 3, 3, 0, 3, 9, 3, 0, 0, 9, 3, 0, 3, 9, 9, 0, 3, 0, 3, 0, 9, 9, 3, 0, 0, 9, 0, 0, 3, 27, 3, 0, 9, 9, 9, 0, 3, 9, 9, 0, 3, 27, 3, 0, 0, 9, 3, 0, 0, 0, 9, 0, 3, 0, 9, 0, 9, 9, 3, 0, 3, 9, 0, 0, 9, 27, 3, 0, 9, 27, 3, 0, 3, 9, 0, 0, 9, 27, 3, 0, 0, 9, 3, 0, 9, 9, 9, 0, 3, 0, 9, 0, 9, 9, 9, 0, 3, 0, 0, 0

Offset: 1

Vaclav Kotesovec, Aug 20 2021

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

Cf. A008966, A074823, A165824.
Cf. A013928, A069201, A048691.
Cf. A001620, A082633.

Table[MoebiusMu[n]^2 * 3^PrimeNu[n], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, (1 + 3*X))[n], ", "))

for(n=1, 100, print1(direuler(p=2, n, (1 - 6*X^2 + 8*X^3 - 3*X^4)/(1 - X)^3)[n], ", "))

Dirichlet g.f.: zeta(s)^3 * Product_{primes p} (1 - 6/p^(2*s) + 8/p^(3*s) - 3/p^(4*s)).
Let f(s) = Product_{primes p} (1 - 6/p^(2*s) + 8/p^(3*s) - 3/p^(4*s)), then Sum_{k=1..n} a(k) ~ n * (f(1)*log(n)^2/2 + log(n)*((3*gamma - 1)*f(1) + f'(1)) + f(1)*(1 - 3*gamma + 3*gamma^2 - 3*sg1) + (3*gamma - 1)*f'(1) + f''(1)/2), where f(1) = Product_{primes p} (1 - 6/p^2 + 8/p^3 - 3/p^4) = 0.1148840440802287887292512767015990978487135526872830176248484270625666728..., f'(1) = f(1) * Sum_{primes p} 12*log(p) / ((p-1)*(p+3)) = 0.5497153490016133577871571904347511299324572220423331992393596243955677299..., f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} (-24*p*(p-1) * log(p)^2 / ((p-1)^2 * (p+3)^2)) = 0.9028322988288094236586622799305270026576436536391185119652318723470259904... and gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633).
a(n) = A008966(n) * A048691(n). - Enrique Pérez Herrero, Oct 27 2022
Multiplicative with a(p) = 3, and a(p^e) = 0 for e >= 2. - Amiram Eldar, Dec 25 2022

A351348 Dirichlet g.f.: Product_{p prime} (1 + 2p^(-s)) / (1 - p^(-s) - p^(-2s)).

Original entry on oeis.org

1, 3, 3, 4, 3, 9, 3, 7, 4, 9, 3, 12, 3, 9, 9, 11, 3, 12, 3, 12, 9, 9, 3, 21, 4, 9, 7, 12, 3, 27, 3, 18, 9, 9, 9, 16, 3, 9, 9, 21, 3, 27, 3, 12, 12, 9, 3, 33, 4, 12, 9, 12, 3, 21, 9, 21, 9, 9, 3, 36, 3, 9, 12, 29, 9, 27, 3, 12, 9, 27, 3, 28, 3, 9, 12, 12, 9, 27, 3, 33, 11, 9, 3, 36, 9, 9, 9

Offset: 1

Ilya Gutkovskiy, Feb 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10^8 terms)

Cf. A000204, A074823, A351219, A351346, A351347.

f[p_, e_] := LucasL[e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 87}]

for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X - X^2))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2022

Multiplicative with a(p^e) = Lucas(e+1).
a(n) = Sum_{d|n} A074823(d) * A351219(n/d).
From Vaclav Kotesovec, Feb 12 2022: (Start)
Let f(s) = Product_{p prime} (1 + 1/(p^(2*s) - p^s - 1)) * (1 - 3/p^(2*s) + 2/p^(3*s)), then
Sum_{k=1..n} a(k) ~ n * (f(1)*log(n)^2/2 + ((3*g-1)*f(1) + f'(1))*log(n) + (1 - 3*g + 3*g^2 - 3*sg1)*f(1) + (3*g-1)*f'(1) + f''(1)/2), where
f(1) = Product_{prime p} (p-1)^3 * (p+2) / (p^2 (p^2 - p - 1)) = 0.76679494740111861346654669603448358442373234633770198438779408968851774...,
f'(1) = f(1) * Sum_{p prime} (4*p^2 - 9*p - 4) * log(p) / (p^4 - 4*p^2 + p + 2) = -0.2518173642312369311596467494348076414732211832249275289370643712012051...,
f''(1) = f'(1)^2/f(1) + f(1) * Sum_{p prime} -p*(8*p^5 - 27*p^4 - 16*p^3 + 32*p^2 + 16*p + 14) * log(p)^2 / (p^4 - 4*p^2 + p + 2)^2 = 4.28643633804365513728313780779157573071314496047204449783182235740130206...,
gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

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

A349923 Dirichlet g.f.: Product_{k>=2} (1 + 2 * k^(-s)).

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 6, 2, 6, 2, 10, 2, 6, 6, 6, 2, 10, 2, 10, 6, 6, 2, 22, 2, 6, 6, 10, 2, 22, 2, 10, 6, 6, 6, 22, 2, 6, 6, 22, 2, 22, 2, 10, 10, 6, 2, 34, 2, 10, 6, 10, 2, 22, 6, 22, 6, 6, 2, 46, 2, 6, 10, 18, 6, 22, 2, 10, 6, 22, 2, 46, 2, 6, 10

Offset: 1

Ilya Gutkovskiy, Dec 05 2021

nonn

Cf. A032302, A045778, A074823, A328706, A349924.

cp's OEIS Frontend

A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).

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A065473 Decimal expansion of the strongly carefree constant: Product_{p prime} (1 - (3*p-2)/(p^3)).

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A074816 a(n) = 3^A001221(n) = 3^omega(n).

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A226177 a(n) = mu(n)*d(n), where mu(n) = A008683 and d(n) = A000005.

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A347149 Dirichlet g.f.: Product_{primes p} (1 + 3/p^s).

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A351348 Dirichlet g.f.: Product_{p prime} (1 + 2*p^(-s)) / (1 - p^(-s) - p^(-2*s)).

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A351348 Dirichlet g.f.: Product_{p prime} (1 + 2p^(-s)) / (1 - p^(-s) - p^(-2s)).