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

Author

Reinhard Zumkeller, May 05 2013

Keywords

• nonn

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

Crossrefs

Cf. A013929, A225245, A225354 (complement).

Programs

Haskell

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

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 *)