A359833 Dirichlet inverse of A359832, where A359832(n) = 1 if the 2-adic valuation of n is either 0 or odd, otherwise 0.

1, -1, -1, 1, -1, 1, -1, -2, 0, 1, -1, -1, -1, 1, 1, 3, -1, 0, -1, -1, 1, 1, -1, 2, 0, 1, 0, -1, -1, -1, -1, -5, 1, 1, 1, 0, -1, 1, 1, 2, -1, -1, -1, -1, 0, 1, -1, -3, 0, 0, 1, -1, -1, 0, 1, 2, 1, 1, -1, 1, -1, 1, 0, 8, 1, -1, -1, -1, 1, -1, -1, 0, -1, 1, 0, -1, 1, -1, -1, -3, 0, 1, -1, 1, 1, 1, 1, 2, -1, 0, 1, -1, 1, 1, 1, 5, -1, 0, 0, 0, -1, -1, -1, 2, -1

Offset

1,8

Comments

Multiplicative because A359832 is.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A359832(n/d) * a(d).

Multiplicative with a(2^e) = (-1)^e*Fibonacci(e), and for p > 2, a(p^e) = -1 if e = 1 and 0 otherwise. - Amiram Eldar, Jan 24 2023

Mathematica

f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := (-1)^e*Fibonacci[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 24 2023 *)

Prog

(PARI) A359833(n) = { my(f=factor(n)); prod(k=1, #f~, if(2==f[k, 1], ((-1)^f[k, 2])*fibonacci(f[k, 2]), -(1==f[k, 2]))); }; \\ Antti Karttunen, Jan 25 2023

Crossrefs

Cf. A000045, A007814, A359832, A359834 (parity of terms).

Sequence in context: A298602 A203949 A070200 * A025914 A284977 A025916

Adjacent sequences: A359830 A359831 A359832 * A359834 A359835 A359836

Keyword

sign,mult

Author

Antti Karttunen, Jan 24 2023

Status

approved