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

Author

Karl-Heinz Hofmann, May 30 2025

Keywords

• nonn
• base
• look
• easy

Examples

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

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

Crossrefs

Cf. A000040, A052548, A178910, A293212.

Programs

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 *)

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

Python

from sympy import primefactors
def A384441(n):
    result = n
    for pf in primefactors(n): result ^= pf
    return result

Formula

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

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

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