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

Author

Antti Karttunen, Feb 10 2016

Keywords

• nonn

Links

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

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).

Programs

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 *)

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

Scheme

(define (A268386 n) (A193231 (A268387 n)))

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)