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A256942
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Number of odd squarefree numbers <= 2^n.
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1
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1, 1, 2, 4, 7, 13, 26, 52, 105, 209, 415, 830, 1661, 3321, 6641, 13279, 26565, 53123, 106237, 212488, 424973, 849945, 1699889, 3399761, 6799540, 13599124, 27198203, 54396423, 108792774, 217585510, 435171212, 870342371, 1740684723, 3481369358, 6962738693, 13925477442
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OFFSET
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0,3
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COMMENTS
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Number of oddly squarefree (A122132) numbers in each new tier > 2^(n-1). - Travis Scott, Jan 14 2023
a(n) is also the number of even squarefree numbers <= 2^(n+1). - Amiram Eldar, Feb 20 2023
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LINKS
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FORMULA
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a(n) = Sum_{j=0..n} (-1)^j*A143658(n-j).
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EXAMPLE
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For n=4 there are 7 odd squarefree numbers <= 2^4, namely 1,3,5,7,11,13,15.
For oddly squarefree we have 2^3 < 10,11,12,13,14,15,16 <= 2^4.
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MAPLE
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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:
a:= n -> add(numtheory:-mobius(i)*g(floor(2^n/i^2)), i=1..floor(2^(n/2))):
seq(a(n), n=0..32);
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MATHEMATICA
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A143658[n_] := Sum[MoebiusMu[i] Floor[2^n/i^2], {i, 1, 2^(n/2)}];
a[n_] := Sum[(-1)^j A143658[n-j], {j, 0, n}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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