A268387 Bitwise-XOR of the exponents of primes in the prime factorization of n.

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

Offset: 1

Author

Antti Karttunen, Feb 05 2016

Keywords

• nonn
• base

Comments

The sums of the first 10^k terms, for k = 1, 2, ..., are 11, 139, 1427, 14207, 141970, 1418563, 14183505, 141834204, 1418330298, 14183245181, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.4183... . - Amiram Eldar, Sep 10 2022

Links

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

Crossrefs

A003987, A028234, A059897 and A067029 are used to express relationships between sequence terms.

Cf. A268390 (indices of zeros).

Sequences with similar definitions: A267115, A267116.

Differs from A136566 for the first time at n=24, where a(24) = 2, while A136566(24) = 4.

Cf. A019565, A052330.

Programs

Mathematica

Table[BitXor @@ Map[Last, FactorInteger@ n], {n, 120}] (* Michael De Vlieger, Feb 12 2016 *)

PARI

a(n) = {my(f = factor(n)); my(b = 0); for (k=1, #f~, b = bitxor(b, f[k,2]);); b;} \\ Michel Marcus, Feb 06 2016

Python

from functools import reduce
from operator import xor
from sympy import factorint
def A268387(n): return reduce(xor,factorint(n).values(),0) # Chai Wah Wu, Aug 31 2022

Formula

a(1) = 0; for n > 1: a(n) = A067029(n) XOR a(A028234(n)). [Here XOR stands for bitwise exclusive-or, A003987.]

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

a(n) <= A267116(n) <= A001222(n).

From Peter Munn, Dec 02 2019 with XOR used as above: (Start)

Defined by: a(p^k) = k, for prime p; a(A059897(n,k)) = a(n) XOR a(k).

a(A052330(n XOR k)) = a(A052330(n)) XOR a(A052330(k)).

a(A019565(n XOR k)) = a(A019565(n)) XOR a(A019565(k)).

(End)