OFFSET
0,2
COMMENTS
Number of squarefree numbers with binary expansion of length n, or with n bits. The sum of these numbers is given by A373123. - Gus Wiseman, Jun 02 2024
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..63 (calculated from the b-file at A143658)
FORMULA
a(n) = Sum_{j=0..-1+2^n} abs(mu(2^n + j)).
a(n)/2^n approaches 1/zeta(2), so limiting sequence is floor(2^n/zeta(2)), n >= 0. - Wouter Meeussen, May 25 2003
EXAMPLE
For n=4: among the 16 numbers of {16, ..., 31}, nine are squarefree [17, 19, 21, 22, 23, 26, 29, 30, 31], so a(4) = 9.
MATHEMATICA
Table[Apply[Plus, Table[Abs[MoebiusMu[2^w+j]], {j, 0, 2^w-1}]], {w, 0, 15}]
(* second program *)
Length/@Split[IntegerLength[Select[Range[10000], SquareFreeQ], 2]]//Most (* Gus Wiseman, Jun 02 2024 *)
PROG
(PARI) { a(n) = sum(m=1, sqrtint(2^(n+1)-1), moebius(m) * ((2^(n+1)-1)\m^2 - (2^n-1)\m^2) ) } \\ Max Alekseyev, Oct 18 2008
CROSSREFS
Partial sums (except first term) are A143658.
Run-lengths of A372475.
Row-sums are A373123.
A053797 gives nonempty lengths of exclusive gaps between squarefree numbers.
For primes between powers of 2:
- sum A293697
For squarefree numbers between primes:
- sum A373197
- min A000040
KEYWORD
nonn
AUTHOR
Labos Elemer, Nov 14 2002
EXTENSIONS
More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 12 2003
More terms from Wouter Meeussen, May 25 2003
a(25)-a(32) from Max Alekseyev, Oct 18 2008
a(33)-a(34) from Amiram Eldar, Jul 17 2024
STATUS
approved