A252233 Characteristic function for the integers that are the product of an odd number of primes each with multiplicity one.

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

Offset

1

Comments

This sequence is the characteristic function for the integers in A030059.

The cumulative sums of the sequence at a(10^k) for k = 1, 2, ..., 6 are 4, 30, 303, 3053, 30421, 303857.

References

P. J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 227, Exercise 5.9.

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

Dirichlet g.f.: (zeta(s)/zeta(2*s) - 1/zeta(s))/2

a(n) = (A008966(n) - A008683(n))/2.

a(n) = 1 if n is of the form p_1*p_2*...*p_r for some odd number r, otherwise a(n) = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/Pi^2 (A104141). - Amiram Eldar, Jul 24 2022

Example

a(4) = 0 because 4 = 2^2 (the prime factors of n must not have exponents other than 1).
a(30) = 1 because 30 = 2*3*5 (there are an odd number of prime factors).

Mathematica

Table[(Abs[MoebiusMu[n]] - MoebiusMu[n])/2, {n, 1, 100}]
a[n_] := If[MoebiusMu[n] == -1, 1, 0]; Array[a, 100] (* Amiram Eldar, Jul 24 2022 *)
onpQ[n_]:=Module[{c=PrimeNu[n]}, OddQ[c]&&c==PrimeOmega[n]]; Table[If[onpQ[n], 1, 0], {n, 100}] (* Harvey P. Dale, Apr 08 2023 *)

Prog

(PARI) A252233(n) = ((issquarefree(n)-moebius(n))/2); \\ Antti Karttunen, Oct 08 2017

Crossrefs

Cf. A092248, A030059, A008966, A008683, A104141, A353629.

Sequence in context: A129950 A010051 A358751 * A374130 A374466 A283991

Adjacent sequences: A252230 A252231 A252232 * A252234 A252235 A252236

Keyword

nonn

Author

Geoffrey Critzer, Mar 21 2015

Status

approved