A178910 Binary XOR of divisors of n.

1, 3, 2, 7, 4, 6, 6, 15, 11, 12, 10, 14, 12, 10, 8, 31, 16, 29, 18, 28, 16, 30, 22, 30, 29, 20, 16, 18, 28, 24, 30, 63, 40, 48, 32, 49, 36, 54, 40, 60, 40, 48, 42, 54, 44, 58, 46, 62, 55, 39, 32, 36, 52, 48, 56, 34, 40, 36, 58, 56, 60, 34, 38, 127, 72, 120, 66, 112, 80, 96, 70

Offset: 1

Author

Franklin T. Adams-Watters, Jun 22 2010

Keywords

• base
• nonn
• look

Comments

If 2^k <= n < 2^(k+1), then also 2^k <= a(n) < 2^(k+1), since any proper divisor of n is < 2^k.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..8191

Crossrefs

Cf. A000203, A178908, A178911, A003987.

Cf. A027750, A072594; subsequences A028982 (odd), A028982 (even).

Programs

Haskell

import Data.Bits (xor)
a178910 = foldl1 xor . a027750_row :: Integer -> Integer
-- Reinhard Zumkeller, Nov 17 2012

PARI

a(n)=local(ds,r);ds=divisors(n);for(k=1,#ds,r=bitxor(r,ds[k]));r

Python

from sympy import divisors
def A178910(n):
    res = 1
    for divisor in divisors(n)[1:]: res ^= divisor
    return res # Karl-Heinz Hofmann, May 30 2025