A369188 - OEIS

A369188

Number of squarefree triangular divisors of n.

1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 3, 1, 1, 3, 1, 1, 3, 1, 2, 3, 1, 1, 3, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 4, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 5, 1, 1, 3, 1, 1, 4, 1, 1, 2, 2, 1, 3, 1, 1, 3, 1, 1, 4, 1, 2, 2, 1, 1, 4, 1, 1, 2, 1, 1, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Inverse Möbius transform of mu(n)^2 * c(n), where c(n) is the characteristic function of triangular numbers (A010054). - Wesley Ivan Hurt, Jun 21 2024

LINKS

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

FORMULA

a(n) = Sum_{d|n} mu(d)^2 * c(d), where c = A010054.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/A061304(k) = 1.83695021... . - Amiram Eldar, Jan 20 2024

MATHEMATICA

Table[Sum[MoebiusMu[d]^2 (Floor[Sqrt[2 d + 1] + 1/2] - Floor[Sqrt[2 d] + 1/2]), {d, Divisors[n]}], {n, 100}]

a[n_] := DivisorSum[n, 1 &, IntegerQ@ Sqrt[8*# + 1] && SquareFreeQ[#] &]; Array[a, 100] (* Amiram Eldar, Jan 20 2024 *)

PROG

(PARI) a(n) = sumdiv(n, d, issquare(8*d+1) && issquarefree(d)); \\ Amiram Eldar, Jan 20 2024

CROSSREFS

Cf. A008683 (mu), A010054, A061304 (squarefree triangular numbers), A369189.

Sequence in context: A275699 A305633 A214123 * A007862 A285851 A055169

Adjacent sequences: A369185 A369186 A369187 * A369189 A369190 A369191

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Jan 15 2024

STATUS

approved