OFFSET
1,2
COMMENTS
If n is written in base-11 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/11)); a(n*11^m)=n^m*11^(m(m+1)/2)*a(n).
a(k*11^m)=k^(m+1)*11^(m(m+1)/2), for 0<k<11.
Asymptotic behavior: a(n)=O(n^((1+log_11(n))/2)); this follows from the inequalities below.
a(n)<=b(n), where b(n)=n^(1+floor(log_11(n)))/p^((1+floor(log_11(n)))*floor(log_11(n))/2); equality holds for n=k*11^m, 0<k<11, m>=0. b(n) can also be written n^(1+floor(log_11(n)))/11^A000217(floor(log_11(n))).
Also: a(n)<=3^((1-log_11(3))/2)*n^((1+log_11(n))/2)=1.346673852...^((1-log_11(3))/2)*11^A000217(log_11(n)), equality holds for n=3*11^m, m>=0.
a(n)>c*b(n), where c=0.4751041275076031053975644472... (see constant A132265).
Also: a(n)>c*(sqrt(2)/2^log_11(sqrt(2)))*n^((1+log_11(n))/2)=0.607848303...*11^00217(log_11(n)).
lim inf a(n)/b(n)=0.4751041275076031053975644472..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_p(n))/2)=0.4751041275076031...*sqrt(2)/2^log_11(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_p(n))/2)=sqrt(3)/3^log_11(sqrt(3))=1.346673852..., for n-->oo.
lim inf a(n)/a(n+1)=0.4751041275076031053975644472... for n-->oo (see constant A132265).
EXAMPLE
a(50)=floor(50/11^0)*floor(50/11^1)=50*4=200; a(63)=315 since 63=58(base-11) and so a(63)=58*5(base-11)=63*5=315.
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Hieronymus Fischer, Aug 20 2007
STATUS
approved