A284556 - OEIS

%I #15 Apr 06 2017 21:21:45

%S 0,0,1,1,0,2,1,1,1,2,1,3,1,2,2,2,0,3,2,3,2,4,2,4,1,3,3,4,1,4,2,2,1,3,

%T 2,5,2,5,4,5,1,6,4,6,3,6,3,5,1,4,4,6,2,7,4,5,2,5,3,6,2,4,3,3,0,4,3,5,

%U 3,7,4,7,2,7,6,9,3,9,5,6,2,7,5,10,4,10,7,9,2,9,6,9,4,8,4,6,1,5,5,8,3,10,6,8,3,9,6,11,4,9,6,7,1,7,5,8,4,9,5

%N Sequence c of the mutual recurrence pair: b(0) = 0, b(1) = 1, b(2n) = c(n), b(2n+1) = b(n) + b(n+1), c(0) = c(1) = 0, c(2n) = b(n), c(2n+1) = c(n) + c(n+1).

%H Antti Karttunen, <a href="/A284556/b284556.txt">Table of n, a(n) for n = 0..8192</a>

%F a(n) = A001222(A284554(n)).

%F Other identities. For all n >= 1:

%F a(n) = (A002487(n) - A102283(n))/2.

%F a(n) = A002487(n) - A000360(n-1).

%F A000360(n-1) - a(n) = A102283(n) ≡ n (mod 3).

%t a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[PrimeOmega[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p + 1, 2])^e) &@ a@ n], {n, 0, 118}] (* or *)

%t a[n_] := Which[n < 2, n, EvenQ@ n, a[n/2], True, a[(n - 1)/2] + a[(n + 1)/2]]; Table[(a[n] - JacobiSymbol[n, 3])/2, {n, 0, 118}] (* _Michael De Vlieger_, Apr 05 2017, after _Alonso del Arte_ at A102283 *)

%o (Scheme, with memoization-macro definec)

%o (definec (A284556 n) (cond ((<= n 1) 0) ((even? n) (A000360with_prep_0 (/ n 2))) (else (+ (A284556 (/ (- n 1) 2)) (A284556 (/ (+ n 1) 2))))))

%o (definec (A000360with_prep_0 n) (cond ((<= n 1) n) ((even? n) (A284556 (/ n 2))) (else (+ (A000360with_prep_0 (/ (- n 1) 2)) (A000360with_prep_0 (/ (+ n 1) 2))))))

%Y Cf. A000360, A001222, A002487, A102283, A284554, A284566 (odd bisection).

%K nonn

%O 0,6

%A _Antti Karttunen_, Apr 05 2017