A355689 Dirichlet inverse of A166486, characteristic function of numbers that are not multiples of 4.

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

Offset

1

Comments

From Antti Karttunen, Dec 31 2022: (Start)

Note the correspondences between four sequences:

A355689 --- abs ---> A353627

^ ^

| |

inv inv

| |

v v

A166486 <--- abs --- A358839

Here inv means that the sequences are Dirichlet Inverses of each other, and abs means taking absolute values.

(End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A166486(n/d) * a(d).

From Antti Karttunen, Dec 22, Dec 31 2022: (Start)

Multiplicative with a(2^e) = (-1)^e, and for odd primes p, a(p^e) = -1 if e = 1, otherwise 0.

For all e >= 0, a(2^e) = A008836(2^e).

For all n >= 0, a(2n+1) = A008683(2n+1).

a(n) = A359156(n) - A359158(n).

(End)

Dirichlet g.f.: 4^s/((4^s-1)*zeta(s)). - Amiram Eldar, Dec 30 2022

Mathematica

f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := (-1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 30 2022 *)

Prog

(PARI)
A166486(n) = !!(n%4);
memoA355689 = Map();
A355689(n) = if(1==n, 1, my(v); if(mapisdefined(memoA355689, n, &v), v, v = -sumdiv(n, d, if(d<n, A166486(n/d)*A355689(d), 0)); mapput(memoA355689, n, v); (v)));
(PARI) A355689(n) = { my(f = factor(n)); prod(k=1, #f~, if(2==f[k, 1], (-1)^f[k, 2], -(1==f[k, 2]))); }; \\ Antti Karttunen, Dec 22 2022

Crossrefs

Cf. A008683, A008836, A166486 (Dirichlet inverse), A353627 (absolute values), A358839 (their Dirichlet inverse), A359157 (positions of the positive terms), A359159 (of the negative terms), A359156, A359158.

Cf. also A087003, A355688, A355690.

Sequence in context: A340374 A070887 A209635 * A353627 A353628 A359551

Adjacent sequences: A355686 A355687 A355688 * A355690 A355691 A355692

Keyword

sign,mult

Author

Antti Karttunen, Jul 15 2022

Status

approved