1,2

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).

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)=sqr(3)/3^log_8(sqr(3))=1.295758534..., for n-->oo.

lim inf a(n)/a(n+1)=0.46456888368647639098... for n-->oo (see constant A132024).

a(70)=floor(70/8^0)*floor(70/8^1)*floor(70/8^2)=70*8*1=560; a(75)=113 since 75=113(base-8) and so

a(75)=113*11*1(base-8)=75*9*1=675.

Cf. A048651, A132024, A132036, A000217.

For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.

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), A132263(p=11), A132264(p=12).

For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Sequence in context: A023769 A023796 A032950 this_sequence A055645 A043320 A044917

Adjacent sequences: A132029 A132030 A132031 this_sequence A132033 A132034 A132035

nonn

Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 20 2007