A072591 In prime factorization of n replace multiplication with bitwise logical 'and'.

1, 2, 3, 2, 5, 2, 7, 2, 3, 0, 11, 2, 13, 2, 1, 2, 17, 2, 19, 0, 3, 2, 23, 2, 5, 0, 3, 2, 29, 0, 31, 2, 3, 0, 5, 2, 37, 2, 1, 0, 41, 2, 43, 2, 1, 2, 47, 2, 7, 0, 1, 0, 53, 2, 1, 2, 3, 0, 59, 0, 61, 2, 3, 2, 5, 2, 67, 0, 3, 0, 71, 2, 73, 0, 1, 2, 3, 0, 79, 0, 3, 0, 83, 2, 1, 2, 1, 2, 89, 0, 5, 2, 3, 2

Offset: 1

Author

Reinhard Zumkeller, Jun 23 2002

Keywords

• nonn

Comments

a(n) = a(A007947(n)); n is prime iff a(n)=n;

a(n)=0 iff n is even and one prime factor is of form 4*k+1.

Examples

			a(35) = a(5*7) = a(5) 'and' a(7) = '101' and '111' = '101' = 5.
		

Links

• Paul Tek, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A072593, A072594, A072592.

Cf. A027746.

Programs

Haskell

import Data.Bits ((.&.))
a072591 = foldl1 (.&.) . a027746_row  -- Reinhard Zumkeller, Jul 05 2013

Mathematica

a[n_] := BitAnd @@ FactorInteger[n][[All, 1]];
Array[a, 100] (* Jean-François Alcover, Nov 16 2021 *)

Python

from sympy import factorint
from operator import _and_
from functools import reduce
def a(n): return reduce(_and_, (f for f in factorint(n))) if n > 1 else 1
print([a(n) for n in range(1, 95)]) # Michael S. Branicky, May 31 2025