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A132269 Product{k>=0, 1+floor(n/2^k)}. 16
1, 2, 6, 8, 30, 36, 56, 64, 270, 300, 396, 432, 728, 784, 960, 1024, 4590, 4860, 5700, 6000, 8316, 8712, 9936, 10368, 18200, 18928, 21168, 21952, 27840, 28800, 31744, 32768, 151470, 156060, 170100, 174960, 210900, 216600, 234000, 240000, 340956, 349272, 374616 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

If n is written in base-2 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product (1+d(m)d(m-1)d(m-2)...d(2)d(1)d(0))*(1+d(m)d(m-1)d(m-2)...d(2)d(1))*(1+d(m)d(m-1)d(m-2)...d(2))*...*(1+d(m)d(m-1)d(m-2))*(1+d(m)d(m-1))*(1+d(m)).

From  Gary W. Adamson, Aug 25 2016: (Start)

Given the following production matrix M =

  1, 0, 0, 0, 0,...

  2, 0, 0, 0, 0,...

  0, 3, 0, 0, 0,...

  0, 4, 0, 0, 0,...

  0, 0, 5, 0, 0,...

  0, 0, 6, 0, 0,...

  0, 0, 0, 7, 0,...

  ...

the sequence is the left-shifted vector as the limit {n..inf} M^n. (End)

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

FORMULA

Recurrence: a(n)=(1+n)*a(floor(n/2)); a(2n)=(1+2n)*a(n); a(n*2^m)=product{1<=k<=m, 1+n*2^k}*a(n).

a(2^m-1)=2^(m(m+1)/2), a(2^m)=2^(m(m+1)/2)*product{0<=k<=m, 1+1/2^k}, m>=1.

a(n)=A132270(2n)=(1+n)*A132270(n).

Asymptotic behavior: a(n)=O(n^((1+log_2(n))/2)); this follows from the inequalities below.

a(n)<=A098844(n)*product{0<=k<=floor(log_2(n)), 1+1/2^k}.

a(n)>=A098844(n)/product{1<=k<=floor(log_2(n)), 1-1/2^k}.

a(n)<c*n^((1+log_2(n))/2)=c*2^A000217(log_2(n)), where c=product{k>=0, 1+1/2^k}=4.7684620580627... (see constant A081845).

a(n)>n^((1+log_2(n))/2)=2^A000217(log_2(n)),

lim sup a(n)/A098844(n)=product{k>=0, 1+1/2^k}=4.7684620580627..., for n-->oo (see constant A081845).

lim inf a(n)/A098844(n)=1/product{k>0, 1-1/2^k}=1/0.288788095086602421..., for n-->oo (see constant A048651).

lim inf a(n)/n^((1+log_2(n))/2)=1, for n-->oo.

lim sup a(n)/n^((1+log_2(n))/2)=product{k>=0, 1+1/2^k}=4.7684620580627..., for n-->oo (see constant A081845).

lim inf a(n+1)/a(n)=product{k>=0, 1+1/2^k}=4.7684620580627... for n-->oo (see constant A081845).

G.f. g(x) satisfies g(x) = (1+2x) g(x^2) + 2 x^2 (1+x) g'(x^2). - Robert Israel, Aug 26 2016

EXAMPLE

a(10)=(1+floor(10/2^0))*(1+floor(10/2^1))*(1+floor(10/2^2))*(1+floor(10/2^3))=11*6*3*2=396; a(17)=4860 since

17=10001(base-2) and so a(17)=(1+10001)*(1+1000)*(1+100)*(1+10)*(1+1)(base-2)=18*9*5*3*2=4860.

MAPLE

f:= proc(n) option remember; (1+n)*procname(floor(n/2)) end proc:

f(0):= 1:

map(f, [$0..100]); # Robert Israel, Aug 26 2016

MATHEMATICA

Table[Product[1 + Floor[2 n/2^k], {k, 2 n}], {n, 0, 42}] (* or *)

Table[Function[w, Times @@ Map[1 + FromDigits[PadRight[w, #], 2] &, Range@ Length@ w]]@ IntegerDigits[n, 2], {n, 0, 42}] (* Michael De Vlieger, Aug 26 2016 *)

PROG

(MAGMA) [1] cat [n le 1 select 2 else (1+n)*Self(Floor(n/2)): n in [1..50]]; // Vincenzo Librandi, Aug 26 2016

CROSSREFS

Cf. A048651, A081845, A132270, A132271(p=10), A132272, A132327(p=3), A132328.

For formulas regarding a general parameter p (i.e. terms 1+floor(n/p^k)) see A132271.

For the product of terms floor(n/p^k) see A098844, A067080, A132027-A132033, A132263, A132264.

Sequence in context: A290249 A321471 A216762 * A053287 A086323 A075999

Adjacent sequences:  A132266 A132267 A132268 * A132270 A132271 A132272

KEYWORD

nonn

AUTHOR

Hieronymus Fischer, Aug 20 2007

STATUS

approved

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Last modified May 30 05:35 EDT 2020. Contains 334712 sequences. (Running on oeis4.)