A268387 - OEIS

%I #36 Sep 10 2022 07:34:29

%S 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,

%T 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,

%U 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

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

%C 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

%H Antti Karttunen, <a href="/A268387/b268387.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

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

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

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

%F From _Peter Munn_, Dec 02 2019 with XOR used as above: (Start)

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

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

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

%F (End)

%t Table[BitXor @@ Map[Last, FactorInteger@ n], {n, 120}] (* _Michael De Vlieger_, Feb 12 2016 *)

%o (Scheme, with memoization-macro definec)

%o (definec (A268387 n) (cond ((= 1 n) 0) (else (A003987bi (A067029 n) (A268387 (A028234 n)))))) ;; A003987bi implements bitwise-xor (see A003987).

%o (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

%o (Python)

%o from functools import reduce

%o from operator import xor

%o from sympy import factorint

%o def A268387(n): return reduce(xor,factorint(n).values(),0) # _Chai Wah Wu_, Aug 31 2022

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

%Y Cf. A268390 (indices of zeros).

%Y Sequences with similar definitions: A267115, A267116.

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

%Y Cf. A019565, A052330.

%K nonn,base

%O 1,4

%A _Antti Karttunen_, Feb 05 2016