A384667 Characteristic function of powers of squarefree numbers (A072774).

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

Offset

1

Comments

a(n) = 1 if and only if all the exponents in the prime factorization of n are equal.

a(n) depends only on the prime signature of n.

Links

Friedjof Tellkamp, Table of n, a(n) for n = 1..10000

Formula

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{n=1..m} a(n) = 6/Pi^2 (A059956).

Dirichlet g.f.: 1 + Sum_{k>=1} (zeta(k*s)/zeta(2*k*s)-1) for s > 1.

a(n) = 1 - chi(A059404)(n), where chi(f) denotes the characteristic function of f. - Peter Luschny, Jun 17 2025

Maple

a := n -> ifelse(1 < nops({seq(p[2], p in ifactors(n)[2])}), 0, 1):
seq(a(n), n = 1..86);  # Peter Luschny, Jun 10 2025

Mathematica

a[n_] := Boole[Equal@@FactorInteger[n][[All, 2]]]; Array[a, 86]

Prog

(PARI) a(n) = n == 1 || #Set(factor(n)[, 2]) == 1; \\ Amiram Eldar, Jun 10 2025
(Python)
from sympy import factorint
def A384667(n):
    e = list(factorint(n).values())
    return int(all(e[0] == a for a in e)) # Chai Wah Wu, Jun 10 2025

Crossrefs

Cf. A005117, A059404, A059956, A072774, A382883.

Sequence in context: A123927 A382883 A089829 * A294935 A242902 A196368

Adjacent sequences: A384664 A384665 A384666 * A384668 A384669 A384670

Keyword

nonn

Author

Friedjof Tellkamp, Jun 07 2025

Status

approved