A293212 Binary XOR of prime divisors of n.

2, 3, 2, 5, 1, 7, 2, 3, 7, 11, 1, 13, 5, 6, 2, 17, 1, 19, 7, 4, 9, 23, 1, 5, 15, 3, 5, 29, 4, 31, 2, 8, 19, 2, 1, 37, 17, 14, 7, 41, 6, 43, 9, 6, 21, 47, 1, 7, 7, 18, 15, 53, 1, 14, 5, 16, 31, 59, 4, 61, 29, 4, 2, 8, 10, 67, 19, 20, 0, 71, 1, 73, 39, 6, 17

Offset: 2

Author

Alex Ratushnyak, Feb 04 2018

Keywords

• nonn
• base

Comments

The sequence of indices of zeros begins: 70, 140, 280, 350, 490, 560, 646, 700, 980, 1120, 1292, 1400, 1750, 1798, 1960, 2145.

Examples

			a(6) = a(24) = 2 XOR 3 = 1.
a(2145) = 3 XOR 5 XOR 11 XOR 13 = 0.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..20000

Crossrefs

Cf. A000040, A072594, A178910.

Programs

Maple

a:= proc(n) local d, r; r:=0; for d in numtheory
      [factorset](n) do r:= Bits[Xor](r, d) od; r
    end:
seq(a(n), n=2..100);  # Alois P. Heinz, Mar 09 2018

PARI

a(n) = my(vp = factor(n)[,1]~, k=0); for (i=1, #vp, k = bitxor(k, vp[i])); k; \\ Michel Marcus, Feb 05 2018

Python

from functools import reduce
from operator import xor
from sympy import primefactors
def A293212(n): return reduce(xor,primefactors(n)) # Chai Wah Wu, Jun 03 2025

Formula

a(n) = n iff n is a prime.