A359579 Dirichlet inverse of A336923, where A336923(n) = 1 if sigma(2n) - sigma(n) is a power of 2, otherwise 0.

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

Offset

1

Links

Antti Karttunen, Table of n, a(n) for n = 1..107163

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A336923(n/d) * a(d).

Multiplicative with a(2^e) = -1 if e = 1, and 0 if e > 1, and for primes p > 2, a(p^e) = (-A036987(p))^e. - Corrected by Amiram Eldar and Antti Karttunen, Jan 24 2023

For all n >= 1, abs(a(A056652(n))) = abs(a(2*A056652(n))) = 1.

For all n >= 1, abs(a(A219174(n))) = 1 if A219174(n) is not a multiple of 4.

Mathematica

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 *)

Prog

(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])); };

Crossrefs

Cf. A036987, A056652, A219174, A336923.

Cf. also A359578.

Sequence in context: A355946 A123272 A266623 * A246142 A091219 A304109

Adjacent sequences: A359576 A359577 A359578 * A359580 A359581 A359582

Keyword

sign,mult

Author

Antti Karttunen, Jan 08 2023

Status

approved