A132028 Product{0<=k<=floor(log_4(n)), floor(n/4^k)}, n>=1.

1, 2, 3, 4, 5, 6, 7, 16, 18, 20, 22, 36, 39, 42, 45, 64, 68, 72, 76, 100, 105, 110, 115, 144, 150, 156, 162, 196, 203, 210, 217, 512, 528, 544, 560, 648, 666, 684, 702, 800, 820, 840, 860, 968, 990, 1012, 1034, 1728, 1764, 1800, 1836, 2028, 2067, 2106, 2145, 2352

Offset

1,2

Comments

If n is written in base-4 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).

Links

Table of n, a(n) for n=1..56.

Formula

Recurrence: a(n)=n*a(floor(n/4)); a(n*4^m)=n^m*4^(m(m+1)/2)*a(n).

a(k*4^m)=k^(m+1)*4^(m(m+1)/2), for 0<k<4.

Asymptotic behavior: a(n)=O(n^((1+log_4(n))/2)); this follows from the inequalities below.

a(n)<=b(n), where b(n)=n^(1+floor(log_4(n)))/4^((1+floor(log_4(n)))*floor(log_4(n))/2); equality holds for n=k*4^m, 0<k<4, m>=0. b(n) can also be written n^(1+floor(log_4(n)))/4^A000217(floor(log_4(n))).

Also: a(n)<=2^(1/4)*n^((1+log_4(n))/2)=1.189207...*4^A000217(log_4(n)), equality holds for n=2*4^m, m>=0.

a(n)>c*b(n), where c=0.4194224417951075977... (see constant A132020).

Also: a(n)>c*2^(1/4)*n^((1+log_4(n))/2)=0.498780...*4^A000217(log_4(n)).

lim inf a(n)/b(n)=0.4194224417951075977..., for n-->oo.

lim sup a(n)/b(n)=1, for n-->oo.

lim inf a(n)/n^((1+log_4(n))/2)=0.4194224417951075977...*2^(1/4), for n-->oo.

lim sup a(n)/n^((1+log_4(n))/2)=2^(1/4), for n-->oo.

lim inf a(n)/a(n+1)=0.4194224417951075977... for n-->oo (see constant A132020).

Example

a(26)=floor(26/4^0)*floor(26/4^1)*floor(26/4^2)=26*6*1=156; a(34)=544 since 34=202(base-4) and so
a(34)=202*20*2(base-4)=34*8*2=544.

Crossrefs

Cf. A048651, A132020, A100221, 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: A162231 A370957 A250045 * A332535 A285724 A193551

Adjacent sequences: A132025 A132026 A132027 * A132029 A132030 A132031

Keyword

nonn

Author

Hieronymus Fischer, Aug 20 2007

Status

approved