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A369188
Number of squarefree triangular divisors of n.
2
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
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
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
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Jan 15 2024
STATUS
approved