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A050354 Number of ordered factorizations of n with one level of parentheses. 4
1, 1, 1, 3, 1, 5, 1, 9, 3, 5, 1, 21, 1, 5, 5, 27, 1, 21, 1, 21, 5, 5, 1, 81, 3, 5, 9, 21, 1, 37, 1, 81, 5, 5, 5, 111, 1, 5, 5, 81, 1, 37, 1, 21, 21, 5, 1, 297, 3, 21, 5, 21, 1, 81, 5, 81, 5, 5, 1, 201, 1, 5, 21, 243, 5, 37, 1, 21, 5, 37, 1, 513, 1, 5, 21, 21, 5, 37, 1, 297, 27, 5, 1, 201 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

Dirichlet inverse of (A074206*A153881). - Mats Granvik, Jan 12 2009

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

FORMULA

Dirichlet g.f.: (2-zeta(s))/(3-2*zeta(s)).

Recurrence for number of ordered factorizations of n with k-1 levels of parentheses is a(n) = k*Sum_{d|n, d<n} a(d), n>1, a(1)= 1/k. - Vladeta Jovovic, May 25 2005

a(p^k) = 3^(k-1).

a(A002110(n)) = A050351(n).

Sum_{k=1..n} a(k) ~ -n^r / (4*r*Zeta'(r)), where r = 2.185285451787482231198145140899733642292971552057774261555354324536... is the root of the equation Zeta(r) = 3/2. - Vaclav Kotesovec, Feb 02 2019

EXAMPLE

For n=6, we have (6) = (3*2) = (2*3) = (3)*(2) = (2)*(3), thus a(6) = 5.

MATHEMATICA

A[n_]:=If[n==1, n/2, 2*Sum[If[d<n, A[d], 0], {d, Divisors[n]}]]; Table[If[n==1, n, A[n]], {n, 1, 100}] (* Indranil Ghosh, May 19 2017 *)

PROG

(PARI)

A050354aux(n) = if(1==n, n/2, 2*sumdiv(n, d, if(d<n, A050354aux(d), 0)));

A050354(n) = if(1==n, n, A050354aux(n)); \\ Antti Karttunen, May 19 2017, after Jovovic's general recurrence.

(Python)

def A(n): return n/2 if n==1 else 2*sum([A(d) for d in divisors(n) if d<n])

def a(n): return n if n==1 else A(n)

print [a(n) for n in xrange(1, 101)] # Indranil Ghosh, May 19 2017, after Antti Karttunen's PARI program

CROSSREFS

Cf. A002033, A050351, A050352, A050353, A050355, A050356, A050357, A050358, A050359.

Sequence in context: A240535 A262397 A155912 * A146434 A126213 A146935

Adjacent sequences:  A050351 A050352 A050353 * A050355 A050356 A050357

KEYWORD

nonn

AUTHOR

Christian G. Bower, Oct 15 1999

EXTENSIONS

Duplicate comment removed by R. J. Mathar, Jul 15 2010

STATUS

approved

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Last modified February 20 21:44 EST 2019. Contains 320362 sequences. (Running on oeis4.)