cp's OEIS Frontend

A132027 a(n) = Product_{k=0..floor(log_3(n))} floor(n/3^k), n>=1.

%I A132027 #9 May 29 2016 16:15:22
%S A132027 1,2,3,4,5,12,14,16,27,30,33,48,52,56,75,80,85,216,228,240,294,308,
%T A132027 322,384,400,416,729,756,783,900,930,960,1089,1122,1155,1728,1776,
%U A132027 1824,2028,2080,2132,2352,2408,2464,3375,3450,3525,3840,3920,4000,4335
%N A132027 a(n) = Product_{k=0..floor(log_3(n))} floor(n/3^k), n>=1.
%C A132027 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 d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m= -2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).
%H A132027 Ivan Neretin, <a href="/A132027/b132027.txt">Table of n, a(n) for n = 1..10000</a>
%F A132027 Recurrence: a(n)=n*a(floor(n/3)); a(n*3^m)=n^m*3^(m(m+1)/2)*a(n).
%F A132027 a(k*3^m)=k^(m+1)*3^(m(m+1)/2), for k=1 or 2.
%F A132027 a(n)<=b(n), where b(n)=n^(1+floor(log_3(n)))/3^(1/2*(1+floor(log_3(n)))*floor(log_3(n))); equality holds if n is a power of 3 or two times a power of 3.
%F A132027 Also: a(n)<=2^((1-log_3(2))/2)*n^((1+log_3(n))/2)=1.1364507...*3^A000217(log_3(n)), equality for n=2*3^m, m>=0.
%F A132027 a(n)>c*b(n), where c=0.3826631966790330232889550... (see constant A132019).
%F A132027 Also: a(n)>c*2^((1-log_3(2))/2)*n^((1+log_3(n))/2)=0.434877...*3^A000217(log_3(n)).
%F A132027 lim inf a(n)/b(n)=0.3826631966790330232889550..., for n-->oo.
%F A132027 lim sup a(n)/b(n)=1, for n-->oo.
%F A132027 lim inf a(n)/n^((1+log_3(n))/2)=0.3826631966790330232889550...*sqrt(2)/2^log_3(sqrt(2)), for n-->oo.
%F A132027 lim sup a(n)/n^((1+log_3(n))/2)=sqrt(2)/2^log_3(sqrt(2)), for n-->oo.
%F A132027 lim inf a(n)/a(n+1)=0.3826631966790330232889550... for n-->oo (see constant A132019).
%F A132027 a(n)=O(n^((1+log_3(n))/2)).
%e A132027 a(11)=floor(11/3^0)*floor(11/3^1)*floor(11/3^2)=11*3*1=33;
%e A132027 a(13)=52 since 13=111(base-3) and so a(13)=111*11*1(base-3)=13*4*1=52.
%t A132027 Table[(f = If[# < 3, #, #*f[Quotient[#, 3]]] &)[n], {n, 51}] (* _Ivan Neretin_, May 29 2016 *)
%Y A132027 Cf. A048651, A098844, A067080, A132019, A100220, A000217.
%Y A132027 For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
%Y A132027 For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132028(p=4)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
%Y A132027 For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.
%K A132027 nonn
%O A132027 1,2
%A A132027 _Hieronymus Fischer_, Aug 13 2007, Aug 20 2007