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A256942 Number of odd squarefree numbers <= 2^n.

%I A256942 #33 Oct 17 2024 17:45:25
%S A256942 1,1,2,4,7,13,26,52,105,209,415,830,1661,3321,6641,13279,26565,53123,
%T A256942 106237,212488,424973,849945,1699889,3399761,6799540,13599124,
%U A256942 27198203,54396423,108792774,217585510,435171212,870342371,1740684723,3481369358,6962738693,13925477442
%N A256942 Number of odd squarefree numbers <= 2^n.
%C A256942 Number of oddly squarefree (A122132) numbers in each new tier > 2^(n-1). - _Travis Scott_, Jan 14 2023
%C A256942 a(n) is also the number of even squarefree numbers <= 2^(n+1). - _Amiram Eldar_, Feb 20 2023
%H A256942 Chai Wah Wu, <a href="/A256942/b256942.txt">Table of n, a(n) for n = 0..73</a> (terms 0..64 from Amiram Eldar)
%F A256942 a(n) = Sum_{j=0..n} (-1)^j*A143658(n-j).
%F A256942 a(n) = (2/3) * A143658(n) + (1/3) * Sum_{i=1..floor(2^(n/2))} A008683(i)*A065359(floor(2^n/i^2)).
%F A256942 a(n) + a(n+1) = A143658(n+1).
%F A256942 a(n) ~ 2^(n+2)/Pi^2. - _Amiram Eldar_, Feb 20 2023
%e A256942 For n=4 there are 7 odd squarefree numbers <= 2^4, namely 1,3,5,7,11,13,15.
%e A256942 For oddly squarefree we have 2^3 < 10,11,12,13,14,15,16 <= 2^4.
%p A256942 g:= proc(n) option remember; local L ; L := convert(n, base, 2) ; (2*n - add( L[i]*(-1)^i, i=1..nops(L)))/3 ; end proc:
%p A256942 a:= n -> add(numtheory:-mobius(i)*g(floor(2^n/i^2)),i=1..floor(2^(n/2))):
%p A256942 seq(a(n),n=0..32);
%t A256942 A143658[n_] := Sum[MoebiusMu[i] Floor[2^n/i^2], {i, 1, 2^(n/2)}];
%t A256942 a[n_] := Sum[(-1)^j A143658[n-j], {j, 0, n}];
%t A256942 Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Sep 22 2022 *)
%Y A256942 Cf. A008683, A039956, A056911, A065359, A122132, A143658.
%K A256942 nonn
%O A256942 0,3
%A A256942 _Robert Israel_, Apr 13 2015
%E A256942 a(33)-a(35) from _Amiram Eldar_, Feb 20 2023