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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. 48
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. J. Mathar, Survey of Dirichlet Series of Multiplicative Arithmetic Functions, arXiv:1106.4038 [math.NT], 2011-2012. See eq. (2.12).
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
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
Sequence in context: A318316 A328721 A165872 * A318312 A318474 A326306
KEYWORD
easy,nonn,mult
AUTHOR
Henry Bottomley, May 29 2001
STATUS
approved

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Last modified May 23 22:02 EDT 2024. Contains 372765 sequences. (Running on oeis4.)