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A061142 Replace each prime factor of n with 2: a(n) = 2^bigomega(n), where bigomega = A001222, number of prime factors counted with multiplicity. 27
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

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

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

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A000079, A001222, A001316, A034444, A123667, A124508, A156552.

Sequence in context: A085191 A188581 A165872 * A278525 A226083 A182730

Adjacent sequences:  A061139 A061140 A061141 * A061143 A061144 A061145

KEYWORD

easy,nonn,mult

AUTHOR

Henry Bottomley, May 29 2001

STATUS

approved

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Last modified November 20 08:12 EST 2017. Contains 294962 sequences.