cp's OEIS Frontend

A260728 Bitwise-OR of the exponents of all 4k+3 primes in the prime factorization of n.

%I A260728 #31 Jun 28 2022 15:28:47
%S A260728 0,0,0,1,0,0,1,1,0,2,0,1,1,0,1,1,0,0,2,1,0,1,1,1,1,0,0,3,1,0,1,1,0,1,
%T A260728 0,1,2,0,1,1,0,0,1,1,1,2,1,1,1,2,0,1,0,0,3,1,1,1,0,1,1,0,1,3,0,0,1,1,
%U A260728 0,1,1,1,2,0,0,1,1,1,1,1,0,4,0,1,1,0,1,1,1,0,2,1,1,1,1,1,1,0,2,3,0,0,1,1,0,1,0,1,3,0,1,1,1,0,1,1,0,2,1,1,1
%N A260728 Bitwise-OR of the exponents of all 4k+3 primes in the prime factorization of n.
%C A260728 A001481 (numbers that are the sum of 2 squares) gives the positions of even terms in this sequence, while its complement A022544 (numbers that are not the sum of 2 squares) gives the positions of odd terms.
%C A260728 If instead of bitwise-oring (A003986) we added in ordinary way the exponents of 4k+3 primes together, we would get the sequence A065339. For the positions where these two sequences differ see A260730.
%H A260728 Antti Karttunen, <a href="/A260728/b260728.txt">Table of n, a(n) for n = 0..10000</a>
%F A260728 If n < 3, a(n) = 0; thereafter, for any even n: a(n) = a(n/2), for any n with its smallest prime factor (A020639) of the form 4k+1: a(n) = a(A032742(n)), otherwise [when A020639(n) is of the form 4k+3] a(n) = A003986(A067029(n),a(A028234(n))).
%F A260728 Other identities. For all n >= 0:
%F A260728 A229062(n) = 1 - A000035(a(n)). [Reduced modulo 2 and complemented, the sequence gives the characteristic function of A001481.]
%F A260728 a(n) = a(A097706(n)). [The result depends only on the prime factors of the form 4k+3.]
%F A260728 a(n) = A267116(A097706(n)).
%F A260728 a(n) = A267113(A267099(n)).
%e A260728 For n = 21 = 3^1 * 7^1 we compute A003986(1,1) = 1, thus a(21) = 1.
%e A260728 For n = 63 = 3^2 * 7^1 we compute A003986(2,1) = A003986(1,2) = 3, thus a(63) = 3.
%t A260728 Table[BitOr @@ (Map[Last, FactorInteger@ n /. {p_, _} /; MemberQ[{0, 1, 2}, Mod[p, 4]] -> Nothing]), {n, 0, 120}] (* _Michael De Vlieger_, Feb 07 2016 *)
%o A260728 (Scheme) (define (A260728 n) (cond ((< n 3) 0) ((even? n) (A260728 (/ n 2))) ((= 1 (modulo (A020639 n) 4)) (A260728 (A032742 n))) (else (A003986bi (A067029 n) (A260728 (A028234 n)))))) ;; A003986bi implements bitwise-or (see A003986).
%o A260728 (Python)
%o A260728 from functools import reduce
%o A260728 from operator import or_
%o A260728 from sympy import factorint
%o A260728 def A260728(n): return reduce(or_,(e for p, e in factorint(n).items() if p & 3 == 3),0) # _Chai Wah Wu_, Jun 28 2022
%Y A260728 Cf. A000035, A001481, A022544, A003986, A020639, A028234, A032742, A067029, A097706, A229062, A260730.
%Y A260728 Cf. also A267113, A267116, A267099.
%Y A260728 Differs from A065339 for the first time at n=21, where a(21) = 1, while A065339(21)=2.
%K A260728 nonn
%O A260728 0,10
%A A260728 _Antti Karttunen_, Aug 12 2015