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A077643
Number of squarefree integers in closed interval [2^n, -1 + 2*2^n], i.e., among 2^n consecutive numbers beginning with 2^n.
28
1, 2, 3, 5, 9, 19, 39, 79, 157, 310, 621, 1246, 2491, 4980, 9958, 19924, 39844, 79672, 159365, 318736, 637457, 1274916, 2549816, 5099651, 10199363, 20398663, 40797299, 81594571, 163189087, 326378438, 652756861, 1305513511, 2611026987, 5222053970, 10444108084
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.
The minimum is A372683, delta A373125, indices A372540.
The maximum is A372889 (except at n=1), delta A373126, indices A143658.
Row-sums are A373123.
A005117 lists squarefree numbers, first differences A076259.
A053797 gives nonempty lengths of exclusive gaps between squarefree numbers.
A029837 counts bits, row-lengths of A030190 and A030308.
For primes between powers of 2:
- sum A293697
- length A036378 or A162145
- min A104080 or A014210, delta A092131, indices A372684
- max A014234, delta A013603, indices A007053
For squarefree numbers between primes:
- sum A373197
- length A373198 = A061398 - 1
- min A000040
- max A112925 (delta A240473), opposite A112926 (delta A240474)
Cf. A010036, A029931, A035100, A049093-A049096, A372473 (firsts of A372472), A372541 (firsts of A372433).
Sequence in context: A119002 A058770 A049910 * A123389 A113984 A110542
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