A345371 Number of squarefree divisors of n whose square does not divide n.

0, 1, 1, 0, 1, 3, 1, 0, 0, 3, 1, 2, 1, 3, 3, 0, 1, 2, 1, 2, 3, 3, 1, 2, 0, 3, 0, 2, 1, 7, 1, 0, 3, 3, 3, 0, 1, 3, 3, 2, 1, 7, 1, 2, 2, 3, 1, 2, 0, 2, 3, 2, 1, 2, 3, 2, 3, 3, 1, 6, 1, 3, 2, 0, 3, 7, 1, 2, 3, 7, 1, 0, 1, 3, 2, 2, 3, 7, 1, 2, 0, 3, 1, 6, 3, 3, 3, 2, 1, 6, 3, 2, 3

Offset

1,6

Links

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

Formula

a(n) = Sum_{d|n} mu(d)^2 * c(n/d^2), where c(n) = ceiling(n) - floor(n).

a(n) = A034444(n) - A323308(n). - Amiram Eldar, Oct 13 2023

Example

a(30) = Sum_{d|30} mu(d)^2 * c(30/d^2) = 1*0 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 = 7.

Mathematica

a[n_] := DivisorSum[n, 1 &, SquareFreeQ[#] && ! Divisible[n, #^2] &]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

Prog

(PARI) a(n) = {my(e = factor(n)[, 2]); 1 << #e - vecprod(apply(x -> min(x, 2), e)); } \\ Amiram Eldar, Oct 13 2023

Crossrefs

Cf. A008683 (mu), A034444, A323308, A344137.

Sequence in context: A035654 A170846 A085604 * A306268 A354490 A365970

Adjacent sequences: A345368 A345369 A345370 * A345372 A345373 A345374

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jun 16 2021

Status

approved