A072594 In prime factorization of n replace multiplication with bitwise logical 'xor'.

1, 2, 3, 0, 5, 1, 7, 2, 0, 7, 11, 3, 13, 5, 6, 0, 17, 2, 19, 5, 4, 9, 23, 1, 0, 15, 3, 7, 29, 4, 31, 2, 8, 19, 2, 0, 37, 17, 14, 7, 41, 6, 43, 11, 5, 21, 47, 3, 0, 2, 18, 13, 53, 1, 14, 5, 16, 31, 59, 6, 61, 29, 7, 0, 8, 10, 67, 17, 20, 0, 71, 2, 73, 39, 3, 19, 12, 12, 79, 5, 0, 43, 83, 4

Offset

1,2

Comments

n is prime iff a(n)=n;

for primes p, k>0: a(p^k)=p*(k mod 2);

for m>1: a(m^2)=0, see A072595.

a(A127812(n)) = n and a(m) <> n for m < A127812(n).

Links

R. Zumkeller, Table of n, a(n) for n = 1..10000

Example

a(35) = a(5*7) = a(5) 'xor' a(7) = '101' xor '111' = '010' = 2.

Mathematica

a[n_] := BitXor @@ Flatten[ Table[ First[#], {Last[#]} ]& /@ FactorInteger[n] ]; Table[a[n], {n, 1, 84}] (* Jean-François Alcover, Mar 11 2013 *)

Prog

(Haskell)
import Data.Bits (xor)
a072594 = foldl1 xor . a027746_row :: Integer -> Integer
-- Reinhard Zumkeller, Nov 17 2012
(PARI) a(n)=if(n==1, return(1)); my(f=factor(n), t); for(i=1, #f~, if(f[i, 2]%2, t=bitxor(t, f[i, 1]))); t \\ Charles R Greathouse IV, Aug 28 2016
(Python)
from sympy import factorint
from operator import __xor__
from functools import reduce
def a(n): return reduce(__xor__, (f for f in factorint(n, multiple=True))) if n > 1 else 1
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, May 31 2025

Crossrefs

Cf. A072591, A072593, A027746, A178910.

Sequence in context: A339767 A071321 A071322 * A353051 A272591 A339694

Adjacent sequences: A072591 A072592 A072593 * A072595 A072596 A072597

Keyword

nonn,nice

Author

Reinhard Zumkeller, Jun 23 2002

Status

approved