A225353 Numbers having no partition into distinct squarefree divisors.

4, 8, 9, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 63, 64, 68, 72, 75, 76, 80, 81, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 126, 128, 135, 136, 140, 144, 147, 148, 150, 152, 153, 160, 162, 164

Offset

1,1

Comments

A225245(a(n)) = 0.

Verified up to a(366) = 1000, a(n) is also the order of a finite group G for which |Out(G)|>|G| for at least one group of order a(n), Out(G) being the outer automorphism group of G. - Miles Englezou, Apr 19 2024

By definition, a(n) is nonsquarefree for every n, since every squarefree number m has a trivial partition into distinct squarefree divisors m = m. - Miles Englezou, Apr 20 2024

If k is a term then so is k*m where m|k. - David A. Corneth, Apr 27 2024

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Mathematica

f[n_] := Coefficient[Product[If[MoebiusMu[d] != 0, 1 + x^d, 1], {d, Divisors[n]}], x, n];
Select[Range[200], f[#] == 0&] (* Jean-François Alcover, May 04 2024, after Ilya Gutkovskiy in A225245 *)

Prog

(Haskell)
import Data.List (elemIndices)
a225353 n = a225353_list !! (n-1)
a225353_list = map (+ 1) $ elemIndices 0 a225245_list

Crossrefs

Cf. A013929, A225245, A225354 (complement).

Sequence in context: A242663 A160053 A140612 * A034023 A086368 A034024

Adjacent sequences: A225350 A225351 A225352 * A225354 A225355 A225356

Keyword

nonn

Author

Reinhard Zumkeller, May 05 2013

Status

approved