cp's OEIS Frontend

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.

%I A077643 #33 Jul 17 2024 08:58:34
%S A077643 1,2,3,5,9,19,39,79,157,310,621,1246,2491,4980,9958,19924,39844,79672,
%T A077643 159365,318736,637457,1274916,2549816,5099651,10199363,20398663,
%U A077643 40797299,81594571,163189087,326378438,652756861,1305513511,2611026987,5222053970,10444108084
%N 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.
%C A077643 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
%H A077643 Amiram Eldar, <a href="/A077643/b077643.txt">Table of n, a(n) for n = 0..63</a> (calculated from the b-file at A143658)
%F A077643 a(n) = Sum_{j=0..-1+2^n} abs(mu(2^n + j)).
%F A077643 a(n)/2^n approaches 1/zeta(2), so limiting sequence is floor(2^n/zeta(2)), n >= 0. - _Wouter Meeussen_, May 25 2003
%e A077643 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.
%t A077643 Table[Apply[Plus, Table[Abs[MoebiusMu[2^w+j]], {j, 0, 2^w-1}]], {w, 0, 15}]
%t A077643 (* second program *)
%t A077643 Length/@Split[IntegerLength[Select[Range[10000],SquareFreeQ],2]]//Most (* _Gus Wiseman_, Jun 02 2024 *)
%o A077643 (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
%Y A077643 Cf. A077641, A077642.
%Y A077643 Partial sums (except first term) are A143658.
%Y A077643 Run-lengths of A372475.
%Y A077643 The minimum is A372683, delta A373125, indices A372540.
%Y A077643 The maximum is A372889 (except at n=1), delta A373126, indices A143658.
%Y A077643 Row-sums are A373123.
%Y A077643 A005117 lists squarefree numbers, first differences A076259.
%Y A077643 A053797 gives nonempty lengths of exclusive gaps between squarefree numbers.
%Y A077643 A029837 counts bits, row-lengths of A030190 and A030308.
%Y A077643 For primes between powers of 2:
%Y A077643 - sum A293697
%Y A077643 - length A036378 or A162145
%Y A077643 - min A104080 or A014210, delta A092131, indices A372684
%Y A077643 - max A014234, delta A013603, indices A007053
%Y A077643 For squarefree numbers between primes:
%Y A077643 - sum A373197
%Y A077643 - length A373198 = A061398 - 1
%Y A077643 - min A000040
%Y A077643 - max A112925 (delta A240473), opposite A112926 (delta A240474)
%Y A077643 Cf. A010036, A029931, A035100, A049093-A049096, A372473 (firsts of A372472), A372541 (firsts of A372433).
%K A077643 nonn
%O A077643 0,2
%A A077643 _Labos Elemer_, Nov 14 2002
%E A077643 More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 12 2003
%E A077643 More terms from _Wouter Meeussen_, May 25 2003
%E A077643 a(25)-a(32) from _Max Alekseyev_, Oct 18 2008
%E A077643 a(33)-a(34) from _Amiram Eldar_, Jul 17 2024