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%I #37 Jan 01 2023 02:29:39
%S 1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,0,1,0,1,1,1,1,1,1,1,
%T 1,0,1,1,1,1,1,1,1,1,0,1,1,1,0,0,1,1,1,0,1,1,1,1,1,1,1,1,0,1,1,1,1,1,
%U 1,1,1,0,1,1,0,1,1,1,1,1,0,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,0,0,0,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,0,1,1,1,0
%N a(n) = 1 if the odd part of n is squarefree, otherwise 0.
%C Sequence gives the absolute values of A209635. See discussion there.
%C From _Antti Karttunen_, Dec 30 2022: (Start)
%C Note the correspondences between four sequences:
%C A355689 --- abs ---> A353627 (this sequence)
%C ^ ^
%C | |
%C inv inv
%C | |
%C v v
%C A166486 <--- abs --- A358839
%C Here inv means that the sequences are Dirichlet Inverses of each other, and abs means taking absolute values.
%C (End)
%H Antti Karttunen, <a href="/A353627/b353627.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.
%F Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = 1 if e = 1 and 0 if e > 1.
%F a(n) = A008966(A000265(n)).
%F a(n) = abs(A209635(n)) = 1 - A353626(n).
%F a(n) >= A323239(n).
%F a(n) >= A336477(n).
%F a(n) <= A353628(n).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 8/Pi^2 (A217739). - _Amiram Eldar_, Jul 23 2022
%F a(n) = abs(A355689(n)) = A359156(n) + A359158(n). - _Antti Karttunen_, Dec 30 2022
%F Dirichlet g.f.: zeta(s)/(zeta(2*s)*(1-1/4^s)). - _Amiram Eldar_, Jan 01 2023
%t a[n_] := If[SquareFreeQ[n/2^IntegerExponent[n, 2]], 1, 0]; Array[a, 100] (* _Amiram Eldar_, Jul 23 2022 *)
%o (PARI) A353627(n) = issquarefree(n>>valuation(n, 2));
%Y Characteristic function of A122132. Positions of zeros: A038838.
%Y Absolute values of A209635 and A355689, binary complement of A353626.
%Y Parity of A342417, A354347, A354823.
%Y Sum of A359156 and A359158.
%Y Cf. A358839 (Dirichlet inverse), A166486 (their absolute values).
%Y Differs from A353628 at the positions given by A353456.
%Y Cf. A000265, A008966, A217739, A336477.
%Y Cf. also A323239, A359377.
%K nonn,mult
%O 1
%A _Antti Karttunen_, May 02 2022