Search: a353456 -id:a353456
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A353627
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a(n) = 1 if the odd part of n is squarefree, otherwise 0.
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+10
21
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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, 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, 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
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OFFSET
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1
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COMMENTS
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Sequence gives the absolute values of A209635. See discussion there.
Note the correspondences between four sequences:
^ ^
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inv inv
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v v
Here inv means that the sequences are Dirichlet Inverses of each other, and abs means taking absolute values.
(End)
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LINKS
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FORMULA
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Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = 1 if e = 1 and 0 if e > 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 8/Pi^2 (A217739). - Amiram Eldar, Jul 23 2022
Dirichlet g.f.: zeta(s)/(zeta(2*s)*(1-1/4^s)). - Amiram Eldar, Jan 01 2023
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MATHEMATICA
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a[n_] := If[SquareFreeQ[n/2^IntegerExponent[n, 2]], 1, 0]; Array[a, 100] (* Amiram Eldar, Jul 23 2022 *)
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PROG
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(PARI) A353627(n) = issquarefree(n>>valuation(n, 2));
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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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, 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, 1, 1, -1, 0, -1
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OFFSET
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1
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COMMENTS
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The sequence is the Moebius function applied to n after all powers of 2 have been cast out, thus it tells whether n is divisible by a square of any odd prime. Multiplicative with a(p^e) = 1 if p = 2, -1 if p > 2 and e = 1; 0 if p > 2 and e > 1.
There seems to be an interesting connection between this sequence and the reduction of the sequence A001001 modulo 2. We have that |a(n)| = A001001(n) mod 2 for n = 1, 2, ..., 80. This equality does not hold for n in {81, 2*81, 3*81, ..., 7*81, 625, 8*81, 10*81, ...}. - John M. Campbell, Jul 17 2016
a(n) = 0 iff there is a prime p > 2 such that p^2 | n. A001001(n) is even iff there exist a prime p > 2 and e = 4k+2 or 4k+3 such that p^e || n. Therefore, |a(n)| differs from A001001(n) mod 2 iff a(n) = 0 and A001001(n) is odd. That happens iff there is a prime p > 2 such that p^e || n, with e > 1, but every such prime-power factor satisfies e mod 4 in {0, 1}. - Álvar Ibeas, Nov 28 2017
Note added May 03 2022: The absolute values of the terms is now given by A353627, while the parity of the sequence A001001 is given by A353628, and the positions of their differences by A353456, where a PARI-program implements Álvar's above insight.
The average order of Sum_{k=1..n} a(k) ~ (-log(n/Pi)^2 - gamma^2 - 2*sg1 + Pi^2/12) / log(2)^2 + 1/6, where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Feb 03 2019
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LINKS
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FORMULA
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Dirichlet g.f.: 1/( zeta(s)*(1-1/2^s)^2). - R. J. Mathar, Mar 12 2012
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MATHEMATICA
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Table[MoebiusMu[n/2^IntegerExponent[n, 2]], {n, 80}] (* Michael De Vlieger, Jul 18 2016 *)
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PROG
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(PARI) a(n) = moebius(n/2^(valuation(n, 2))); \\ Michel Marcus, Dec 10 2014
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CROSSREFS
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Cf. A353627 gives the absolute values and is the characteristic function of A122132, whose complement A038838 gives the position of zeros here.
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KEYWORD
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mult,sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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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, 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, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 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
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OFFSET
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1
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COMMENTS
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LINKS
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FORMULA
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Multiplicative with a(p^e) = A000035(((p^(e+1)-1)(p^(e+2)-1)) / ((p-1)(p^2-1))). - Antti Karttunen, Dec 20 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/12 = 0.822467... (A072691). - Amiram Eldar, Oct 23 2023
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MATHEMATICA
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f[p_, e_] := Mod[Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 2}], 2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 23 2023 *)
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PROG
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(PARI)
(PARI) A353628(n) = { my(f = factor(n)); prod(k=1, #f~, my(p=f[k, 1], e=f[k, 2]); (((p^(e+1)-1)*(p^(e+2)-1)) / ((p-1)*(p^2-1)))%2); }; \\ Antti Karttunen, Dec 20 2022
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CROSSREFS
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Characteristic function of A353511.
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94
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OFFSET
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1,2
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COMMENTS
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Differs from A122132 (which is a subsequence of this sequence) for the first time by having term 81. Sequence A353456 gives terms that are present here but not in the former. See also the discussion in A209635.
If k is present in the sequence, then 2*k and A000265(k) are present also.
The asymptotic density of this sequence is Pi^2/12 = 0.822467... (A072691). - Amiram Eldar, Oct 23 2023
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LINKS
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MATHEMATICA
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Position[Array[DivisorSum[#, # DivisorSigma[1, #] &] &, 94], _?OddQ][[All, 1]] (* Michael De Vlieger, May 03 2022 *)
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PROG
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(PARI)
A001001(n) = sumdivmult(n, d, sigma(d)*d);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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