A166698 Totally multiplicative sequence with a(p) = a(p-1) - 1 for prime p.

1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1

Offset: 1

Author

Jaroslav Krizek, Oct 18 2009

Keywords

• sign
• mult

Comments

From Antti Karttunen, Dec 30 2022: (Start)

Note the correspondences between four sequences:

A087003 --- abs ---> A323239

^ ^

| |

inv inv

| |

v v

A000035 <--- abs --- A166698 (this sequence)

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..65537

Crossrefs

Cf. A000035 (absolute values), A001222, A003961, A008836, A323239 (Dirichlet inverse).

Cf. A046337 (positions of positive terms), A067019 (of negative terms), A353557, A353558.

Cf. also A358839, A359378.

Programs

PARI

A166698(n) = { my(f = factor(n)); prod(k=1, #f~, if(2==f[k, 1], 0, (-1)^f[k, 2])); }; \\ Antti Karttunen, Dec 19 2022

Formula

Multiplicative with a(p^e) = (a(p-1)-1)^e.

If n = Product p(k)^e(k) then a(n) = Product (a(p(k)-1)-1)^e(k).

Multiplicative with a(p^e) = 0 if p = 2, with a(p^e) = 1 if p > 2 and e is even, with a(p^e) = -1 if p > 2 and e is odd.

a(p) = -1 for prime p > 2.

a(1) = 1, for k >= 1: a(2k) = 0, a(2k - 1) = 1 if A001222(2k - 1) is even, a(2k - 1) = -1 if A001222(2k - 1) is odd, where A001222(n) = bigomega(n).

Sum_{d|n} a(d) * A000012(d) = Sum_{d|n} a(d) * A000012(d/n) = A053866(n) = A093709(n) for n>= 1.

a(n) = A000035(n) * A008836(n). - Antti Karttunen, Sep 14 2017

From Antti Karttunen_, Dec 19 & Dec 30 2022: (Start)

a(A003961(n)) = A008836(n).

a(n) = A353557(n) - A353558(n).

(End)

Extensions

More terms from Antti Karttunen, Sep 14 2017