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A132269 a(n) = Product_{k>=0} (1 + floor(n/2^k)).

%I A132269 #23 Mar 05 2022 01:01:52
%S A132269 1,2,6,8,30,36,56,64,270,300,396,432,728,784,960,1024,4590,4860,5700,
%T A132269 6000,8316,8712,9936,10368,18200,18928,21168,21952,27840,28800,31744,
%U A132269 32768,151470,156060,170100,174960,210900,216600,234000,240000,340956,349272,374616
%N A132269 a(n) = Product_{k>=0} (1 + floor(n/2^k)).
%C A132269 If n is written in base 2 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 (1+d(m)d(m-1)d(m-2)...d(2)d(1)d(0))*(1+d(m)d(m-1)d(m-2)...d(2)d(1))*(1+d(m)d(m-1)d(m-2)...d(2))*...*(1+d(m)d(m-1)d(m-2))*(1+d(m)d(m-1))*(1+d(m)).
%C A132269 From _Gary W. Adamson_, Aug 25 2016: (Start)
%C A132269 Given the following production matrix M =
%C A132269   1, 0, 0, 0, 0, ...
%C A132269   2, 0, 0, 0, 0, ...
%C A132269   0, 3, 0, 0, 0, ...
%C A132269   0, 4, 0, 0, 0, ...
%C A132269   0, 0, 5, 0, 0, ...
%C A132269   0, 0, 6, 0, 0, ...
%C A132269   0, 0, 0, 7, 0, ...
%C A132269   ...
%C A132269 the sequence is the left-shifted vector as lim_{n->infinity} M^n. (End)
%H A132269 Robert Israel, <a href="/A132269/b132269.txt">Table of n, a(n) for n = 0..10000</a>
%F A132269 Recurrence: a(n)=(1+n)*a(floor(n/2)); a(2n)=(1+2n)*a(n); a(n*2^m) = (Product_{k=1..m} (1 + n*2^k))*a(n).
%F A132269 a(2^m-1) = 2^(m*(m+1)/2), a(2^m) = 2^(m*(m+1)/2)*Product_{k=0..m} (1 + 1/2^k), m>=1.
%F A132269 a(n) = A132270(2n) = (1+n)*A132270(n).
%F A132269 Asymptotic behavior: a(n) = O(n^((1+log_2(n))/2)); this follows from the inequalities below.
%F A132269 a(n) <= A098844(n)*Product_{k=0..floor(log_2(n))} (1 + 1/2^k).
%F A132269 a(n) >= A098844(n)/Product_{k=1..floor(log_2(n))} (1 - 1/2^k).
%F A132269 a(n) < c*n^((1+log_2(n))/2) = c*2^A000217(log_2(n)), where c = Product_{k>=0} (1 + 1/2^k) = 4.7684620580627... (see constant A081845).
%F A132269 a(n) > n^((1+log_2(n))/2) = 2^A000217(log_2(n)),
%F A132269 lim sup a(n)/A098844(n) = Product_{k>=0} (1 + 1/2^k) = 4.7684620580627..., for n->oo (see constant A081845).
%F A132269 lim inf a(n)/A098844(n) = 1/Product_{k>=1} (1 - 1/2^k) = 1/0.288788095086602421..., for n->oo (see constant A048651).
%F A132269 lim inf a(n)/n^((1+log_2(n))/2) = 1, for n->oo.
%F A132269 lim sup a(n)/n^((1+log_2(n))/2) = Product_{k>=0} (1 + 1/2^k) = 4.7684620580627..., for n->oo (see constant A081845).
%F A132269 lim inf a(n+1)/a(n) = Product_{k>=0} (1 + 1/2^k) = 4.7684620580627... for n->oo (see constant A081845).
%F A132269 G.f. g(x) satisfies g(x) = (1+2x)*g(x^2) + 2*x^2*(1+x)*g'(x^2). - _Robert Israel_, Aug 26 2016
%e A132269 a(10) = (1 + floor(10/2^0))*(1 + floor(10/2^1))*(1 + floor(10/2^2))*(1 + floor(10/2^3)) = 11*6*3*2 = 396;
%e A132269 a(17) = 4860 since 17 = 10001_2 and so a(17) = (1+10001_2)*(1+1000_2)*(1+100_2)*(1+10_2)*(1+1) = 18*9*5*3*2 = 4860.
%p A132269 f:= proc(n) option remember; (1+n)*procname(floor(n/2)) end proc:
%p A132269 f(0):= 1:
%p A132269 map(f, [$0..100]); # _Robert Israel_, Aug 26 2016
%t A132269 Table[Product[1 + Floor[2 n/2^k], {k, 2 n}], {n, 0, 42}] (* or *)
%t A132269 Table[Function[w, Times @@ Map[1 + FromDigits[PadRight[w, #], 2] &, Range@ Length@ w]]@ IntegerDigits[n, 2], {n, 0, 42}] (* _Michael De Vlieger_, Aug 26 2016 *)
%o A132269 (Magma) [1] cat [n le 1 select 2 else (1+n)*Self(Floor(n/2)): n in [1..50]]; // _Vincenzo Librandi_, Aug 26 2016
%Y A132269 Cf. A048651, A081845, A132270, A132271(p=10), A132272, A132327(p=3), A132328.
%Y A132269 For formulas regarding a general parameter p (i.e., terms 1+floor(n/p^k)) see A132271.
%Y A132269 For the product of terms floor(n/p^k) see A098844, A067080, A132027-A132033, A132263, A132264.
%K A132269 nonn
%O A132269 0,2
%A A132269 _Hieronymus Fischer_, Aug 20 2007