|
%I #7 Oct 27 2014 10:43:22
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,36,38,40,42,44,46,48,50,52,
%T 81,84,87,90,93,96,99,102,105,144,148,152,156,160,164,168,172,176,225,
%U 230,235,240,245,250,255,260,265,324,330,336,342,348,354,360,366,372
%N Product{0<=k<=floor(log_9(n)), floor(n/9^k)}, n>=1.
%C If n is written in base-9 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).
%F Recurrence: a(n)=n*a(floor(n/9)); a(n*9^m)=n^m*9^(m(m+1)/2)*a(n).
%F a(k*9^m)=k^(m+1)*9^(m(m+1)/2), for 0<k<9.
%F Asymptotic behavior: a(n)=O(n^((1+log_9(n))/2)); this follows from the inequalities below.
%F a(n)<=b(n), where b(n)=n^(1+floor(log_9(n)))/9^((1+floor(log_9(n)))*floor(log_9(n))/2); equality holds for n=k*9^m, 0<k<9, m>=0. b(n) can also be written n^(1+floor(log_9(n)))/9^A000217(floor(log_9(n))).
%F Also: a(n)<=3^(1/4)*n^((1+log_9(n))/2)=1.316074013...*9^A000217(log_9(n)), equality holds for n=3*9^m, m>=0.
%F a(n)>c*b(n), where c=0.4689451783670236932832800... (see constant A132024).
%F Also: a(n)>c*2^((1-log_9(2))/2)*n^((1+log_9(n))/2)=0.4689451783670...*1.267747616...*9^A000217(log_9(n)).
%F lim inf a(n)/b(n)=0.4689451783670236932832800..., for n-->oo.
%F lim sup a(n)/b(n)=1, for n-->oo.
%F lim inf a(n)/n^((1+log_9(n))/2)=0.4689451783670236932832800...*sqrt(2)/2^log_9(sqrt(2)), for n-->oo.
%F lim sup a(n)/n^((1+log_9(n))/2)=3^(1/4)=1.316074013..., for n-->oo.
%F lim inf a(n)/a(n+1)=0.4689451783670236932832800... for n-->oo (see constant A132025).
%e a(85)=floor(85/9^0)*floor(85/9^1)*floor(85/9^2)=85*9*1=765; a(88)=792 since 88=107(base-9) and so a(88)=107*10*1(base-9)=88*9*1=792.
%Y Cf. A048651, A132025, A132037, A000217.
%Y For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
%Y For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132032(p=8), A067080(p=10), A132263(p=11), A132264(p=12).
%Y For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.
%K nonn,base
%O 1,2
%A _Hieronymus Fischer_, Aug 20 2007
|
