A384441 Binary XOR of n and the prime factors of n.

1, 0, 0, 6, 0, 7, 0, 10, 10, 13, 0, 13, 0, 11, 9, 18, 0, 19, 0, 19, 17, 31, 0, 25, 28, 21, 24, 25, 0, 26, 0, 34, 41, 49, 33, 37, 0, 55, 41, 47, 0, 44, 0, 37, 43, 59, 0, 49, 54, 53, 33, 59, 0, 55, 57, 61, 41, 37, 0, 56, 0, 35, 59, 66, 73, 72, 0, 87, 81, 70, 0, 73, 0, 109

Offset

1,4

Links

Alois P. Heinz, Table of n, a(n) for n = 1..16384

Karl-Heinz Hofmann, Graph up to n = 2^17

Karl-Heinz Hofmann, Zoom trip of the graph with number of prime factors of n colored.

Wikipedia, Bitwise operation: XOR

Formula

a(n) = XOR(n,A293212(n)).

a(n) = 0 <=> n is prime.

a(2^n) = A052548(n) for n>=2.

Example

For n = 12 the prime factors are {2,3} -> a(12) = 12 XOR 2 XOR 3 = 13.
a(13) = 13 XOR 13 = 0.

Maple

f:= l-> `if`(l=[], 0, Bits[Xor](l[1], f(l[2..-1]))):
a:= n-> f([n, map(i-> i[1], ifactors(n)[2])[]]):
seq(a(n), n=1..74);  # Alois P. Heinz, May 30 2025

Mathematica

a[n_] := BitXor @@ Join[{n}, FactorInteger[n][[;; , 1]]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, May 30 2025 *)

Prog

(Python)
from sympy import primefactors
def A384441(n):
    result = n
    for pf in primefactors(n): result ^= pf
    return result
(PARI) a(n) = my(f=factor(n)[, 1]); my(b=n); for (k=1, #f, b=bitxor(b, f[k])); b; \\ Michel Marcus, May 30 2025

Crossrefs

Cf. A000040, A052548, A178910, A293212.

Sequence in context: A222747 A222608 A353091 * A021900 A351401 A382555

Adjacent sequences: A384438 A384439 A384440 * A384442 A384443 A384444

Keyword

nonn,base,look,easy

Author

Karl-Heinz Hofmann, May 30 2025

Status

approved