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

%I A098844 #24 Jan 15 2025 01:46:34
%S A098844 1,2,3,8,10,18,21,64,72,100,110,216,234,294,315,1024,1088,1296,1368,
%T A098844 2000,2100,2420,2530,5184,5400,6084,6318,8232,8526,9450,9765,32768,
%U A098844 33792,36992,38080,46656,47952,51984,53352,80000,82000,88200,90300
%N A098844 a(1)=1, a(n) = n*a(floor(n/2)).
%H A098844 Hieronymus Fischer, <a href="/A098844/b098844.txt">Table of n, a(n) for n = 1..1000</a>
%F A098844 a(2^n) = 2^(n*(n+1)/2) = A006125(n+1).
%F A098844 From _Hieronymus Fischer_, Aug 13 2007: (Start)
%F A098844 a(n) = Product_{k=0..floor(log_2(n))} floor(n/2^k), n>=1.
%F A098844 Recurrence:
%F A098844 a(n*2^m) = n^m*2^(m(m+1)/2)*a(n).
%F A098844 a(n) <= n^((1+log_2(n))/2) = 2^A000217(log_2(n)); equality iff n is a power of 2.
%F A098844 a(n) >= c(n)*(n+1)^((1 + log_2(n+1))/2) for n != 2,
%F A098844 where c(n) = Product_{k=1..floor(log_2(n))} (1 - 1/2^k); equality holds iff n+1 is a power of 2.
%F A098844 a(n) > c*(n+1)^((1 + log_2(n+1))/2)
%F A098844 where c = 0.288788095086602421... (see constant A048651).
%F A098844 lim inf a(n)/n^((1+log_2(n))/2)=0.288788095086602421... for n-->oo.
%F A098844 lim sup a(n)/n^((1+log_2(n))/2) = 1 for n-->oo.
%F A098844 lim inf a(n)/a(n+1) = 0.288788095086602421... for n-->oo (see constant A048651).
%F A098844 a(n) = O(n^((1+log_2(n))/2)). (End)
%e A098844 a(10) = floor(10/2^0)*floor(10/2^1)*floor(10/2^2)*floor(10/2^3) = 10*5*2*1 = 100;
%e A098844 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.
%t A098844 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 *)
%o A098844 (PARI) a(n)=if(n<2,1,n*a(floor(n/2)))
%o A098844 (Python)
%o A098844 from math import prod
%o A098844 def A098844(n): return n*prod(n//2**k for k in range(1,n.bit_length()-1)) # _Chai Wah Wu_, Jun 07 2022
%Y A098844 Cf. A048651, A067080, A132027, A132028, A132029, A132030, A132019, A132026, A132038.
%Y A098844 For formulas regarding a general parameter p (i.e., terms floor(n/p^k)) see A132264.
%Y A098844 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).
%Y A098844 For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.
%K A098844 nonn
%O A098844 1,2
%A A098844 _Benoit Cloitre_, Nov 03 2004
%E A098844 Formula section edited by _Hieronymus Fischer_, Jun 13 2012