OFFSET
1,2
COMMENTS
If n is written in base-8 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).
FORMULA
Recurrence: a(n)=n*a(floor(n/8)); a(n*8^m)=n^m*8^(m(m+1)/2)*a(n).
a(k*8^m)=k^(m+1)*8^(m(m+1)/2), for 0<k<8.
Asymptotic behavior: a(n)=O(n^((1+log_8(n))/2)); this follows from the inequalities below.
a(n)<=b(n), where b(n)=n^(1+floor(log_8(n)))/8^((1+floor(log_8(n)))*floor(log_8(n))/2); equality holds for n=k*8^m, 0<k<8, m>=0. b(n) can also be written n^(1+floor(log_8(n)))/8^A000217(floor(log_8(n))).
Also: a(n)<=3^((1-log_8(3))/2)*n^((1+log_8(n))/2) = 1.295758534...*8^A000217(log_8(n)), equality holds for n=3*8^m, m>=0.
a(n)>c*b(n), where c = 0.46456888368647639098... (see constant A132024).
Also: a(n)>c*2^(1/3)*n^((1+log_8(n))/2)=0.4645688836...*1.25992105...*8^A000217(log_8(n)).
lim inf a(n)/b(n)=0.46456888368647639098..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_8(n))/2)=0.46456888368647639098...*2^(1/3), for n-->oo.
lim sup a(n)/n^((1+log_8(n))/2)=sqrt(3)/3^log_8(sqrt(3))=1.295758534..., for n-->oo.
lim inf a(n)/a(n+1)=0.46456888368647639098... for n-->oo (see constant A132024).
EXAMPLE
a(70)=floor(70/8^0)*floor(70/8^1)*floor(70/8^2)=70*8*1=560;
For n=75, 75=113(base-8) and so a(75)=113*11*1(base-8)=75*9*1=675.
CROSSREFS
KEYWORD
nonn
AUTHOR
Hieronymus Fischer, Aug 20 2007
STATUS
approved