A351411 Number of divisors of n not of the form p^p, p prime.

1, 2, 2, 2, 2, 4, 2, 3, 3, 4, 2, 5, 2, 4, 4, 4, 2, 6, 2, 5, 4, 4, 2, 7, 3, 4, 3, 5, 2, 8, 2, 5, 4, 4, 4, 8, 2, 4, 4, 7, 2, 8, 2, 5, 6, 4, 2, 9, 3, 6, 4, 5, 2, 7, 4, 7, 4, 4, 2, 11, 2, 4, 6, 6, 4, 8, 2, 5, 4, 8, 2, 11, 2, 4, 6, 5, 4, 8, 2, 9, 4, 4, 2, 11, 4, 4, 4, 7, 2, 12, 4, 5, 4

Offset

1,2

Links

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

Formula

a(n) = tau(n) - Sum_{d|n} [rad(d) = Omega(d)*[omega(d) = 1]], where [ ] is the Iverson bracket.

a(n) = A000005(n) - A129251(n).

Sum_{k=1..n} a(k) ~ n * (log(n) + c), where c = A147533 - A094289 = -0.1329269215... . Amiram Eldar, Oct 01 2023

Example

a(108) = 10; 2 of the 12 divisors of 108 are of the form p^p (p prime), namely 4 = 2^2 and 27 = 3^3; therefore a(108) = 12-2 = 10.

Mathematica

f1[p_, e_] := e + 1; f2[p_, e_] := If[e < p, 0, 1]; a[1] = 1; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Plus @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Oct 01 2023 *)

Prog

(PARI) a(n) = {my(f = factor(n)); vecprod(apply(x -> x+1, f[, 2])) - sum(i = 1, #f~, f[i, 2] >= f[i, 1]); } \\ Amiram Eldar, Oct 01 2023

Crossrefs

Cf. A000005 (tau), A001221 (omega), A001222 (Omega), A007947 (rad).

Cf. A051674, A129251.

Sequence in context: A036555 A046927 A366563 * A084718 A154851 A281854

Adjacent sequences: A351408 A351409 A351410 * A351412 A351413 A351414

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 10 2022

Status

approved