A268386 - OEIS

%I #26 Jun 19 2020 04:21:27

%S 0,1,1,3,1,0,1,2,3,0,1,2,1,0,0,5,1,2,1,2,0,0,1,3,3,0,2,2,1,1,1,4,0,0,

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

%U 0,1,1,1,1,0,2,2,0,1,1,4,5,0,1,3,0,0,0,3,1,3,0,2,0,0,0,5,1,2,2,0,1,1,1,3,1,0,1,1,1,1,0,4,1,1,0,2,2,0,0,2

%N a(n) = A193231(A268387(n)).

%H Antti Karttunen, <a href="/A268386/b268386.txt">Table of n, a(n) for n = 1..16384</a>

%F The following two formulas are equivalent because A193231 distributes over bitwise XOR (A003987):

%F a(n) = A193231(A268387(n)) and

%F a(n) = A268387(A268385(n)).

%F a(2^k) = A193231(k). - _Peter Munn_, May 07 2020

%F From _Peter Munn_, Jun 02 2020: (Start)

%F Alternative definition, for n, k >= 1, where XOR denotes A003987:

%F a(prime(n)) = 1, where prime(n) = A000040(n);

%F a(n^2) = a(n) XOR (2 * a(n)) = A048720(a(n), 3);

%F a(A059897(n, k)) = a(n) XOR a(k).

%F (End)

%t f[n_] := Which[0 <= # <= 1, #, EvenQ@ #, BitXor[2 #, #] &[f[#/2]], True, BitXor[#, 2 # + 1] &[f[(# - 1)/2]]] &@ Abs@ n; {0}~Join~Table[f[BitXor @@ Map[Last, FactorInteger@ n]], {n, 2, 120}] (* _Michael De Vlieger_, Feb 12 2016, after _Robert G. Wilson v_ at A048724 and A065621 *)

%o (Scheme) (define (A268386 n) (A193231 (A268387 n)))

%o (PARI)

%o a268387(n) = {my(f = factor(n), b = 0); for (k=1, #f~, b = bitxor(b, f[k, 2]); ); b; }

%o a193231(n) = {my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2)};

%o a(n) = a193231(a268387(n)); \\ _Michel Marcus_, May 09 2020

%Y A003987, A048720, A059897, A193231, A268385, A268387 are used in definitions of this sequence.

%Y Cf. A000028 (indices of odd numbers), A000379 (indices of even numbers), A268390 (indices of zeros).

%K nonn

%O 1,4

%A _Antti Karttunen_, Feb 10 2016