A373549 a(n) is the parity of the n-th powerful number.

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

Offset

1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Teerapat Srichan, The odd/even dichotomy for the set of square-full numbers, Applied Mathematics E-Notes, Vol. 20 (2020), pp. 528-531.

Wikipedia, Powerful number.

Formula

a(n) = A001694(n) mod 2 = A000035(A001694(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (2 - sqrt(2))/(3 - sqrt(2)) = 0.3693980625... .

Mathematica

Mod[Select[Range[5000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &], 2]

Prog

(PARI) lista(kmax) = for(k = 1, kmax, if(ispowerful(k), print1(k % 2, ", ")));

Crossrefs

Characteristic function of A363189.

Cf. A000035, A001694, A062739, A335851, A363191, A363192, A373550.

Sequence in context: A005171 A283265 A365410 * A181406 A285252 A076404

Adjacent sequences: A373546 A373547 A373548 * A373550 A373551 A373552

Keyword

nonn,easy

Author

Amiram Eldar, Jun 09 2024

Status

approved