A267116 Bitwise-OR of the exponents of primes in the prime factorization of n.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 3, 1, 1, 1, 3, 2, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 5, 2, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 3, 6, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 5, 1, 3, 3, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 3

Offset: 1

Author

Antti Karttunen, Feb 03 2016

Keywords

• nonn

Examples

			For n = 4 = 2^2, bitwise-OR of 2 alone is 2, thus a(4) = 2.
For n = 6 = 2^1 * 3^1, when we take a bitwise-or of 1 and 1, we get 1, thus a(6) = 1.
For n = 24 = 2^3 * 3^1, bitwise-or of 3 and 1 ("11" and "01" in binary) gives "11", thus a(24) = 3.
For n = 210 = 2^1 * 3^1 * 5^1 * 7^1, bitwise-or of 1, 1, 1 and 1 gives 1, thus a(210) = 1.
For n = 720 = 2^4 * 3^2 * 5^1, bitwise-or of 4, 2 and 1 ("100", "10" and "1" in binary) gives 7 ("111" in binary), thus a(720) = 7.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A001222, A003986, A007814, A028234, A067029.

Cf. A000290 (indices of even numbers).

Cf. A000037 (indices of odd numbers).

Nonunit terms of A005117, A062503, A113849 give the positions of ones, twos, fours respectively in this sequence.

Sequences with similar definitions: A260728, A267113, A267115 (bitwise-AND) and A268387 (bitwise-XOR of exponents).

Sequences with related analysis: A267114, A268374, A268375, A268376.

Sequences A088529, A136565 and A181591 coincide with a(n) for n: 2 <= n < 24.

A003961, A059896 are used to express relationship between terms of this sequence.

Related to A087207 via A225546.

Programs

Maple

read("transforms"):
A267116 := proc(n)
    local a,e ;
    a := 0 ;
    for e in ifactors(n)[2] do
        a := ORnos(a,op(2,e)) ;
    end do:
    a ;
end proc: # R. J. Mathar, Feb 16 2021

Mathematica

{0}~Join~Rest@ Array[BitOr @@ Map[Last, FactorInteger@ #] &, 120] (* Michael De Vlieger, Feb 04 2016 *)

PARI

a(n)=my(f = factor(n)); my(b = 0); for (k=1, #f~, b = bitor(b, f[k,2]);); b; \\ Michel Marcus, Feb 05 2016

PARI

a(n)=if(n>1, fold(bitor, factor(n)[,2]), 0) \\ Charles R Greathouse IV, Aug 04 2016

Python

from functools import reduce
from operator import or_
from sympy import factorint
def A267116(n): return reduce(or_,factorint(n).values(),0) # Chai Wah Wu, Aug 31 2022

Formula

a(1) = 0; for n > 1: a(n) = A067029(n) OR a(A028234(n)). [Here OR stands for bitwise-or, A003986.]

Other identities and observations. For all n >= 1:

a(n) = A007814(n) OR A260728(n) OR A267113(n).

a(n) = A001222(n) - A268374(n).

A268387(n) <= a(n) <= A001222(n).

From Peter Munn, Jan 08 2020: (Start)

a(A059896(n,k)) = a(n) OR a(k).

a(A003961(n)) = a(n).

a(n^2) = 2*a(n).

a(n) = A087207(A225546(n)).

a(A225546(n)) = A087207(n).

(End)