A351408 Number of divisors of n that are either trivial or are nonsquares with a square divisor > 1.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 5, 1, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 4, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 2, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 2, 1, 1, 1, 7

Offset

1,8

Links

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

Index entries for sequences computed from exponents in factorization of n

Formula

a(n) = Sum_{d|n} [c(d) = mu(d)^2], where [ ] is the Iverson bracket and c is the characteristic function of squares (A010052).

a(n) = A048105(n) - A046951(n) + 2. - Amiram Eldar, Oct 06 2023

Example

a(96) = 7; 96 has the trivial divisor (=1), and the 6 divisors 8,12,24,32,48,96 which all have a square divisor > 1 but are not themselves square.

Mathematica

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ (e + 1) - 2^Length[e] - Times @@ (1 + Floor[e/2]) + 2]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Oct 06 2023 *)

Prog

(PARI) a(n) = {my(f = factor(n), e = f[, 2], d = numdiv(f), nu = omega(f)); d - 2^nu - vecprod(apply(x -> x\2 + 1, e)) + 2; } \\ Amiram Eldar, Oct 06 2023

Crossrefs

Cf. A008683 (mu), A010052, A046951, A048105.

Sequence in context: A347708 A050330 A339890 * A205788 A214054 A330739

Adjacent sequences: A351405 A351406 A351407 * A351409 A351410 A351411

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 11 2022

Status

approved