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A132328 Product{k>0, 1+floor(n/3^k)}. 16
1, 1, 1, 2, 2, 2, 3, 3, 3, 8, 8, 8, 10, 10, 10, 12, 12, 12, 21, 21, 21, 24, 24, 24, 27, 27, 27, 80, 80, 80, 88, 88, 88, 96, 96, 96, 130, 130, 130, 140, 140, 140, 150, 150, 150, 192, 192, 192, 204, 204, 204, 216, 216, 216, 399, 399, 399, 420, 420, 420, 441, 441, 441, 528 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

If n is written in base-3 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))*(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)).

LINKS

Table of n, a(n) for n=0..63.

FORMULA

Recurrence: a(n)=(1+floor(n/3))*a(floor(n/3)); a(3n)=(1+n)*a(n); a(n*3^m)=product{0<=k<m, 1+n*3^k}*a(n).

a(k*3^m-j)=k^m*3^(m(m-1)/2), for 0<k<3, 0<j<3, m>=1. a(3^m)=p^(m(m-1)/2)*product{0<=k<m, 1+1/3^k}

a(n)=A132327(floor(n/3)=A132327(n)/(1+n).

Asymptotic behavior: a(n)=O(n^((log_3(n)-1)/p)); this follows from the inequalities below.

a(n)<=A132027(n)/(n+1)*product{0<=k<=floor(log_3(n)), 1+1/3^k}.

a(n)>=A132027(n)/((n+1)*product{0<k<=floor(log_3(n)), 1-1/3^k}).

a(n)<c*n^((1+log_3(n))/2)/(n+1)=c*2^A000217(log_3(n))/(n+1), where c=product{k>0, 1+1/3^k}=3.12986803713402307587769821345767... (see constant A132323).

a(n)>n^((1+log_3(n))/2)/(n+1)=3^A000217(log_3(n))/(n+1).

lim sup n*a(n)/A132027(n)=2*product{k>0, 1+1/3^k}=3.12986803713402307587769821345767..., for n-->oo (see constant A132323).

lim inf n*a(n)/A132027(n)=1/product{k>0, 1-1/3^k}=1/0.560126077927948944969792243314140014..., for n-->oo (see constant A100220).

lim inf a(n)/n^((1+log_3(n))/2)=1, for n-->oo.

lim sup a(n)/n^((1+log_3(n))/2)=2*product{k>0, 1+1/3^k}=3.12986803713402307587769821345767..., for n-->oo (see constant A132323).

lim inf a(n+1)/a(n)=2*product{k>0, 1+1/3^k}=3.12986803713402307587769821345767... for n-->oo (see constant A132323).

EXAMPLE

a(12)=(1+floor(12/3^1))*(1+floor(12/3^2))=5*2=10; a(19)=21 since 19=201(base-3) and so a(19)=(1+20)*(1+2)(base-3)=7*3=21.

CROSSREFS

Cf. A100220, A132323, A132027, A132038, A132270(p=2), A132272(p=10).

For formulas regarding a general parameter p (i.e. terms 1+floor(n/p^k)) see A132272.

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

Sequence in context: A029058 A046026 A139801 * A064822 A238337 A104484

Adjacent sequences:  A132325 A132326 A132327 * A132329 A132330 A132331

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Aug 20 2007

STATUS

approved

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Last modified September 20 05:51 EDT 2019. Contains 327212 sequences. (Running on oeis4.)