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A372135 Nonsquarefree numbers not in A225353; equivalently, nonsquarefree numbers in A225354.

%I A372135 #34 May 28 2024 06:46:46
%S A372135 12,60,84,132,156,204,228,276,348,372,420,444,492,516,564,636,660,708,
%T A372135 732,780,804,852,876,924,948,996,1020,1068,1092,1140,1164,1212,1236,
%U A372135 1284,1308,1356,1380,1428,1524,1540,1572,1596,1644,1668,1716,1740,1788,1812,1820
%N A372135 Nonsquarefree numbers not in A225353; equivalently, nonsquarefree numbers in A225354.
%C A372135 Every number in A225353 is nonsquarefree. a(n) corresponds to those numbers which are nonsquarefree yet contain at least one partition into distinct squarefree divisors.
%C A372135 Verified up to a(26) = 996: except for 12, a(n) is also the order of a finite group G for which |Out(G)|<|G| for all isomorphism classes of G where the order of G is nonsquarefree. |Out(G)|<|G| for all isomorphism classes of groups with squarefree order in the same range.
%C A372135 If k is a term, then so is m * k where m is squarefree and coprime to k. - _Robert Israel_, Apr 21 2024
%C A372135 Comparison with other similar sequences:
%C A372135 For values up to and including a(2000)=76044:
%C A372135          b(n):        | 12*A276378|  12*A007310| 12*A038179|  4*A243128|  A357686
%C A372135  --------------------------------------------------------------------------------
%C A372135   # a(n) not in b(n)  |         73|          70|         74|          0|        1
%C A372135   # b(n) not in a(n)  |          0|         186|        188|         69|       69
%C A372135 First a(n) not in b(n)| a(40)=1540|  a(40)=1540|    a(1)=12|     -     |  a(1)=12
%C A372135 First b(n) not in a(n)|     -     | 12*b(9)=300| 12*b(1)=24| 4*b(5)=140| b(4)=140
%F A372135 Equals A013929\A225353 and also A225354\A005117.
%e A372135 12 is a term since 12 = 2^2*3 and 12 = 1 + 2 + 3 + 6.
%p A372135 filter:= proc(n) local P,z,d;
%p A372135   if numtheory:-issqrfree(n) then return false fi;
%p A372135   P:= mul(1+z^d, d = select(numtheory:-issqrfree,numtheory:-divisors(n)));
%p A372135   coeff(P,z,n) > 0
%p A372135 end proc:
%p A372135 select(filter, [$1..2000]); # _Robert Israel_, Apr 21 2024
%t A372135 filter[n_] := Module[{P, z, d},
%t A372135    If[SquareFreeQ[n], Return[False]];
%t A372135    P = Product[1 + z^d, {d, Select[Divisors[n], SquareFreeQ]}];
%t A372135    Coefficient[P, z, n] > 0];
%t A372135 Select[Range[2000], If[filter[#], Print[#]; True, False]&] (* _Jean-François Alcover_, May 28 2024, after _Robert Israel_ *)
%Y A372135 Cf. A005117 (squarefree numbers), A013929 (nonsquarefree numbers), A225353, A225354, A007310, A038179, A243128, A276378, A357686.
%K A372135 nonn
%O A372135 1,1
%A A372135 _Miles Englezou_, Apr 20 2024