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%I #8 Oct 27 2014 10:38:39
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,44,46,48,50,52,
%T 54,56,58,60,62,64,99,102,105,108,111,114,117,120,123,126,129,176,180,
%U 184,188,192,196,200,204,208,212,216,275,280,285,290,295,300,305,310
%N Product{0<=k<=floor(log_11(n)), floor(n/11^k)}, n>=1.
%C 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).
%F Recurrence: a(n)=n*a(floor(n/11)); a(n*11^m)=n^m*11^(m(m+1)/2)*a(n).
%F a(k*11^m)=k^(m+1)*11^(m(m+1)/2), for 0<k<11.
%F Asymptotic behavior: a(n)=O(n^((1+log_11(n))/2)); this follows from the inequalities below.
%F 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))).
%F 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.
%F a(n)>c*b(n), where c=0.4751041275076031053975644472... (see constant A132265).
%F Also: a(n)>c*(sqrt(2)/2^log_11(sqrt(2)))*n^((1+log_11(n))/2)=0.607848303...*11^00217(log_11(n)).
%F lim inf a(n)/b(n)=0.4751041275076031053975644472..., for n-->oo.
%F lim sup a(n)/b(n)=1, for n-->oo.
%F lim inf a(n)/n^((1+log_p(n))/2)=0.4751041275076031...*sqrt(2)/2^log_11(sqrt(2)), for n-->oo.
%F lim sup a(n)/n^((1+log_p(n))/2)=sqrt(3)/3^log_11(sqrt(3))=1.346673852..., for n-->oo.
%F lim inf a(n)/a(n+1)=0.4751041275076031053975644472... for n-->oo (see constant A132265).
%e 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.
%Y Cf. A048651, A132019-A132026, A132265, A132267, 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)-A132033(p=9), A067080(p=10), 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
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