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A260728
Bitwise-OR of the exponents of all 4k+3 primes in the prime factorization of n.
8
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, 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, 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
OFFSET
0,10
COMMENTS
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.
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.
LINKS
FORMULA
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))).
Other identities. For all n >= 0:
A229062(n) = 1 - A000035(a(n)). [Reduced modulo 2 and complemented, the sequence gives the characteristic function of A001481.]
a(n) = a(A097706(n)). [The result depends only on the prime factors of the form 4k+3.]
a(n) = A267116(A097706(n)).
a(n) = A267113(A267099(n)).
EXAMPLE
For n = 21 = 3^1 * 7^1 we compute A003986(1,1) = 1, thus a(21) = 1.
For n = 63 = 3^2 * 7^1 we compute A003986(2,1) = A003986(1,2) = 3, thus a(63) = 3.
MATHEMATICA
Table[BitOr @@ (Map[Last, FactorInteger@ n /. {p_, _} /; MemberQ[{0, 1, 2}, Mod[p, 4]] -> Nothing]), {n, 0, 120}] (* Michael De Vlieger, Feb 07 2016 *)
PROG
(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).
(Python)
from functools import reduce
from operator import or_
from sympy import factorint
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
CROSSREFS
Cf. also A267113, A267116, A267099.
Differs from A065339 for the first time at n=21, where a(21) = 1, while A065339(21)=2.
Sequence in context: A356241 A091430 A362221 * A065339 A122434 A141571
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 12 2015
STATUS
approved