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

1,1

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

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)

Prog

(Scheme, with memoization-macro) (definec (A166698 n) (if (= 1 n) n (* (+ -1 (A166698 (+ -1 (A020639 n)))) (A166698 (A032742 n))))) ;; Antti Karttunen, Sep 14 2017
(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

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.

Sequence in context: A056594 A374134 A374136 * A250299 A361122 A193497

Adjacent sequences: A166695 A166696 A166697 * A166699 A166700 A166701

Keyword

sign,mult

Author

Jaroslav Krizek, Oct 18 2009

Extensions

More terms from Antti Karttunen, Sep 14 2017

Status

approved