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

%I A089608 #25 Jan 04 2023 02:05:15
%S A089608 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,
%T A089608 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,
%U A089608 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
%N A089608 a(n) = ((-1)^(n+1)*A002425(n)) modulo 6.
%C A089608 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).
%H A089608 Antti Karttunen, <a href="/A089608/b089608.txt">Table of n, a(n) for n = 1..65536</a>
%H A089608 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.
%H A089608 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>.
%F A089608 a(n) = 5 - 4*A035263(n).
%F A089608 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 7/3. - _Amiram Eldar_, Nov 28 2022
%F A089608 From _Amiram Eldar_, Jan 04 2023: (Start)
%F A089608 Multiplicative with a(2^e) = 5 if e is odd, and 1 if e is even, a(p^e) = 1 for p >= 3.
%F A089608 Dirichlet g.f.: zeta(s)*(2^s+5)/(2^s+1). (End)
%t A089608 a[n_] := Mod[IntegerExponent[n, 2], 2] * 4 + 1; Array[a, 100] (* _Amiram Eldar_, Nov 28 2022 *)
%o A089608 (PARI) a(n)=numerator(2/n*(4^n-1)*bernfrac(2*n))%6
%o A089608 (PARI) a(n)=valuation(n,2)%2 * 4 + 1; \\ _Andrew Howroyd_, Aug 01 2018
%o A089608 (Scheme)
%o A089608 (define (A035263 n) (let loop ((n n) (i 1)) (cond ((odd? n) (modulo i 2)) (else (loop (/ n 2) (+ 1 i))))))
%o A089608 (define (A089608 n) (- 5 (* 4 (A035263 n))))
%o A089608 ;; _Antti Karttunen_, Sep 11 2017
%Y A089608 Cf. A002425, A035263, A056832.
%K A089608 nonn,mult
%O A089608 1,2
%A A089608 _Benoit Cloitre_, Dec 30 2003
%E A089608 Keyword:mult added by _Andrew Howroyd_, Aug 01 2018