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A225353 Numbers having no partition into distinct squarefree divisors.

%I A225353 #26 May 19 2024 11:57:01
%S A225353 4,8,9,16,18,20,24,25,27,28,32,36,40,44,45,48,49,50,52,54,56,63,64,68,
%T A225353 72,75,76,80,81,88,90,92,96,98,99,100,104,108,112,116,117,120,121,124,
%U A225353 125,126,128,135,136,140,144,147,148,150,152,153,160,162,164
%N A225353 Numbers having no partition into distinct squarefree divisors.
%C A225353 A225245(a(n)) = 0.
%C A225353 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
%C A225353 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
%C A225353 If k is a term then so is k*m where m|k. - _David A. Corneth_, Apr 27 2024
%H A225353 Reinhard Zumkeller, <a href="/A225353/b225353.txt">Table of n, a(n) for n = 1..1000</a>
%t A225353 f[n_] := Coefficient[Product[If[MoebiusMu[d] != 0, 1 + x^d, 1], {d, Divisors[n]}], x, n];
%t A225353 Select[Range[200], f[#] == 0&] (* _Jean-François Alcover_, May 04 2024, after _Ilya Gutkovskiy_ in A225245 *)
%o A225353 (Haskell)
%o A225353 import Data.List (elemIndices)
%o A225353 a225353 n = a225353_list !! (n-1)
%o A225353 a225353_list = map (+ 1) $ elemIndices 0 a225245_list
%Y A225353 Cf. A013929, A225245, A225354 (complement).
%K A225353 nonn
%O A225353 1,1
%A A225353 _Reinhard Zumkeller_, May 05 2013