A268386 a(n) = A193231(A268387(n)).

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, 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, 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

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

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

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

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

a(2^k) = A193231(k). - Peter Munn, May 07 2020

From Peter Munn, Jun 02 2020: (Start)

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

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

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

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

(End)

Mathematica

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 *)

Prog

(Scheme) (define (A268386 n) (A193231 (A268387 n)))
(PARI)
a268387(n) = {my(f = factor(n), b = 0); for (k=1, #f~, b = bitxor(b, f[k, 2]); ); b; }
a193231(n) = {my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2)};
a(n) = a193231(a268387(n)); \\ Michel Marcus, May 09 2020

Crossrefs

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

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

Sequence in context: A291624 A291635 A308243 * A122779 A120323 A320476

Adjacent sequences: A268383 A268384 A268385 * A268387 A268388 A268389

Keyword

nonn

Author

Antti Karttunen, Feb 10 2016

Status

approved