A098844 a(1)=1, a(n) = n*a(floor(n/2)).

1, 2, 3, 8, 10, 18, 21, 64, 72, 100, 110, 216, 234, 294, 315, 1024, 1088, 1296, 1368, 2000, 2100, 2420, 2530, 5184, 5400, 6084, 6318, 8232, 8526, 9450, 9765, 32768, 33792, 36992, 38080, 46656, 47952, 51984, 53352, 80000, 82000, 88200, 90300

Offset: 1

Author

Benoit Cloitre, Nov 03 2004

Keywords

• nonn

Examples

			a(10) = floor(10/2^0)*floor(10/2^1)*floor(10/2^2)*floor(10/2^3) = 10*5*2*1 = 100;
a(17) = 1088 since 17 = 10001(base 2) and so a(17) = 10001*1000*100*10*1(base 2) = 17*8*4*2*1 = 1088.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A048651, A067080, A132027, A132028, A132029, A132030, A132019, A132026, A132038.

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=3 to p=12 see 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.

Programs

Mathematica

lst={};Do[p=n;s=1;While[p>1,p=IntegerPart[p/2];s*=p;];AppendTo[lst,s],{n,1,6!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 28 2009 *)

PARI

a(n)=if(n<2,1,n*a(floor(n/2)))

Python

from math import prod
def A098844(n): return n*prod(n//2**k for k in range(1,n.bit_length()-1)) # Chai Wah Wu, Jun 07 2022

Formula

a(2^n) = 2^(n*(n+1)/2) = A006125(n+1).

From Hieronymus Fischer, Aug 13 2007: (Start)

a(n) = Product_{k=0..floor(log_2(n))} floor(n/2^k), n>=1.

Recurrence:

a(n*2^m) = n^m*2^(m(m+1)/2)*a(n).

a(n) <= n^((1+log_2(n))/2) = 2^A000217(log_2(n)); equality iff n is a power of 2.

a(n) >= c(n)*(n+1)^((1 + log_2(n+1))/2) for n != 2,

where c(n) = Product_{k=1..floor(log_2(n))} (1 - 1/2^k); equality holds iff n+1 is a power of 2.

a(n) > c*(n+1)^((1 + log_2(n+1))/2)

where c = 0.288788095086602421... (see constant A048651).

lim inf a(n)/n^((1+log_2(n))/2)=0.288788095086602421... for n-->oo.

lim sup a(n)/n^((1+log_2(n))/2) = 1 for n-->oo.

lim inf a(n)/a(n+1) = 0.288788095086602421... for n-->oo (see constant A048651).

a(n) = O(n^((1+log_2(n))/2)). (End)

Extensions

Formula section edited by Hieronymus Fischer, Jun 13 2012