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A359579
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Dirichlet inverse of A336923, where A336923(n) = 1 if sigma(2n) - sigma(n) is a power of 2, otherwise 0.
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3
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1, -1, -1, 0, 0, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1
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OFFSET
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1
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LINKS
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FORMULA
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a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A336923(n/d) * a(d).
For all n >= 1, abs(a(A219174(n))) = 1 if A219174(n) is not a multiple of 4.
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MATHEMATICA
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f[p_, e_] := If[2^IntegerExponent[p + 1, 2] == p + 1, (-1)^e, 0]; f[2, e_] := If[e == 1, -1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 24 2023 *)
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PROG
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(PARI)
A209229(n) = (n && !bitand(n, n-1));
A359579(n) = { my(f=factor(n)); prod(k=1, #f~, if(2==f[k, 1], -(1==f[k, 2]), (-A209229(1+f[k, 1]))^f[k, 2])); };
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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