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A356163 a(n) = 1 if sum of prime factors of n (taken with multiplicity) is even, otherwise 0. 10
1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
LINKS
FORMULA
a(n) = A059841(A001414(n)).
a(n) >= A353557(n).
For x, y >= 1, a(x*y) >= a(x) * a(y).
a(x*y) = 1 - a(x) - a(y) + 2*a(x)*a(y). - Robert Israel, Nov 03 2022
a(n) = (1+A359154(n))/2). - Antti Karttunen, Dec 19 2022
a(n) = a(A000265(n)) = A065043(A000265(n)) = A353557(A000265(n)). - Antti Karttunen, Jan 15 2023
MAPLE
f:= proc(n) local t; 1-add(t[1]*t[2], t = ifactors(n)[2]) mod 2 end proc:
map(f, [$1..100]); # Robert Israel, Nov 03 2022
MATHEMATICA
{1}~Join~Array[Boole@ EvenQ[Total[Times @@@ FactorInteger[#]]] &, 119, 2] (* Michael De Vlieger, Nov 02 2022 *)
PROG
(PARI)
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A059841(n) = (1-(n%2));
(Python)
from functools import reduce
from operator import ixor, and_
from itertools import starmap
from sympy import factorint
def A356163(n): return (reduce(ixor, starmap(and_, factorint(n).items()), 0)&1)^1 # Chai Wah Wu, Dec 21 2022
CROSSREFS
Characteristic function of A036349, whose complement A335657 gives the positions of 0's.
Cf. A000265, A001414, A065043 [= a(A003961(n))], A353557, A359154, A359773 (Dirichlet inverse).
Sequence in context: A113430 A214529 A113681 * A295895 A179416 A360845
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 02 2022
STATUS
approved

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Last modified March 28 16:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)