A295316 a(n) = 1 if there are no even exponents in the prime factorization of n, 0 otherwise.

1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1

Offset

1

Comments

Characteristic function for A268335.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions.

Index entries for sequences computed from exponents in factorization of n.

Formula

Multiplicative with a(p^e) = A000035(e).

a(1) = 1, for n > 1, a(n) = A000035(A067029(n)) * a(A028234(n)).

a(n) = 1 iff A162641(n) = 0.

For n > 1, a(n) = A267115(n) mod 2.

a(n) = A355825(A268385(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A065463. - Amiram Eldar, Jul 23 2022

a(n) = A005361(n) mod 2. - Ridouane Oudra, May 12 2025

Mathematica

Array[Boole[AllTrue[FactorInteger[#][[All, -1]], OddQ]] &, 120] (* Michael De Vlieger, Nov 23 2017 *)

Prog

(PARI)
vecproduct(v) = { my(m=1); for(i=1, #v, m *= v[i]); m; };
A295316(n) = vecproduct(apply(e -> (e%2), factorint(n)[, 2]));
(Scheme)
;; With memoization-macro definec.
(definec (A295316 n) (if (= 1 n) n (if (even? (A067029 n)) 0 (A295316 (A028234 n)))))
(Python)
from sympy import factorint
def A295316(n): return int(all(map(lambda m:m&1, factorint(n).values()))) # Chai Wah Wu, Jan 04 2023

Crossrefs

Cf. A000035, A065463, A072587 (positions of zeros), A268335 (of ones), A162641, A267115, A268385, A355825, A355936 (Dirichlet inverse).

Cf. also A033634, A293449, A005361.

Sequence in context: A255887 A359473 A181183 * A014677 A307425 A210826

Adjacent sequences: A295313 A295314 A295315 * A295317 A295318 A295319

Keyword

nonn,mult

Author

Antti Karttunen, Nov 23 2017

Status

approved