A089608 a(n) = ((-1)^(n+1)*A002425(n)) modulo 6.

1, 5, 1, 1, 1, 5, 1, 5, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 5, 1, 5, 1, 1, 1, 5, 1, 5, 1, 5, 1, 1, 1, 5, 1, 5, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 5, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 5, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 5, 1, 5, 1, 1, 1, 5, 1, 5, 1, 5, 1, 1, 1, 5, 1, 5, 1

Offset: 1

Author

Benoit Cloitre, Dec 30 2003

Keywords

• nonn
• mult

Comments

Let S(1)={1} and S(n+1)=S(n)S'(n) where S'(n) is obtained from S(n) by changing last term using the cyclic permutation 1->5->1, then sequence is S(infinity).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65536
• Index entries for sequences related to binary expansion of n.
• Index entries for sequences that are fixed points of mappings.

Crossrefs

Cf. A002425, A035263, A056832.

Programs

Mathematica

a[n_] := Mod[IntegerExponent[n, 2], 2] * 4 + 1; Array[a, 100] (* Amiram Eldar, Nov 28 2022 *)

PARI

a(n)=numerator(2/n*(4^n-1)*bernfrac(2*n))%6

PARI

a(n)=valuation(n,2)%2 * 4 + 1; \\ Andrew Howroyd, Aug 01 2018

Scheme

(define (A035263 n) (let loop ((n n) (i 1)) (cond ((odd? n) (modulo i 2)) (else (loop (/ n 2) (+ 1 i))))))
(define (A089608 n) (- 5 (* 4 (A035263 n))))
;; Antti Karttunen, Sep 11 2017

Formula

a(n) = 5 - 4*A035263(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 7/3. - Amiram Eldar, Nov 28 2022

From Amiram Eldar, Jan 04 2023: (Start)

Multiplicative with a(2^e) = 5 if e is odd, and 1 if e is even, a(p^e) = 1 for p >= 3.

Dirichlet g.f.: zeta(s)*(2^s+5)/(2^s+1). (End)

Extensions

Keyword:mult added by Andrew Howroyd, Aug 01 2018