A061142 Replace each prime factor of n with 2: a(n) = 2^bigomega(n), where bigomega = A001222, number of prime factors counted with multiplicity.

1, 2, 2, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 4, 16, 2, 8, 2, 8, 4, 4, 2, 16, 4, 4, 8, 8, 2, 8, 2, 32, 4, 4, 4, 16, 2, 4, 4, 16, 2, 8, 2, 8, 8, 4, 2, 32, 4, 8, 4, 8, 2, 16, 4, 16, 4, 4, 2, 16, 2, 4, 8, 64, 4, 8, 2, 8, 4, 8, 2, 32, 2, 4, 8, 8, 4, 8, 2, 32, 16, 4, 2, 16, 4, 4, 4, 16, 2, 16, 4, 8, 4, 4, 4

Offset: 1

Author

Henry Bottomley, May 29 2001

Keywords

• easy
• nonn
• mult

Comments

The inverse Möbius transform of A162510. - R. J. Mathar, Feb 09 2011

Examples

			a(100)=16 since 100=2*2*5*5 and so a(100)=2*2*2*2.
		

Links

• R. Zumkeller, Table of n, a(n) for n = 1..10005
• R. J. Mathar, Survey of Dirichlet Series of Multiplicative Arithmetic Functions, arXiv:1106.4038 [math.NT], 2011-2012. See eq. (2.12).
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000079, A001222, A001316, A034444, A069205 (partial sums), A123667, A124508, A156552.

Programs

Maple

with(numtheory): seq(2^bigomega(n),n=1..95);

Mathematica

Table[2^PrimeOmega[n], {n, 1, 95}] (* Jean-François Alcover, Jun 08 2013 *)

PARI

a(n)=direuler(p=1,n,1/(1-2*X))[n] /* Ralf Stephan, Mar 28 2015 */

PARI

a(n) = 2^bigomega(n); \\ Michel Marcus, Aug 08 2017

Formula

a(n) = Sum_{d divides n} 2^(bigomega(d)-omega(d)) = Sum_{d divides n} 2^(A001222(d) - A001221(d)). - Benoit Cloitre, Apr 30 2002

a(n) = A000079(A001222(n)), i.e., a(n)=2^bigomega(n). - Emeric Deutsch, Feb 13 2005

Totally multiplicative with a(p) = 2. - Franklin T. Adams-Watters, Oct 04 2006

Dirichlet g.f.: Product_{p prime} 1/(1-2*p^(-s)). - Ralf Stephan, Mar 28 2015

a(n) = A001316(A156552(n)). - Antti Karttunen, May 29 2017

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} 1/(1 - 1/(p^s - 1)^2). - Vaclav Kotesovec, Mar 14 2023