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Number of squarefree triangular divisors of n.
2

%I #14 Jun 21 2024 17:51:40

%S 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,

%T 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,

%U 2,2,1,3,1,1,3,1,1,4,1,2,2,1,1,4,1,1,2,1,1,5

%N Number of squarefree triangular divisors of n.

%C 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

%H Amiram Eldar, <a href="/A369188/b369188.txt">Table of n, a(n) for n = 1..10000</a>

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

%F 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

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

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

%o (PARI) a(n) = sumdiv(n, d, issquare(8*d+1) && issquarefree(d)); \\ _Amiram Eldar_, Jan 20 2024

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

%K nonn,easy

%O 1,3

%A _Wesley Ivan Hurt_, Jan 15 2024