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Nonsquarefree numbers not in A225353; equivalently, nonsquarefree numbers in A225354.
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%I #34 May 28 2024 06:46:46

%S 12,60,84,132,156,204,228,276,348,372,420,444,492,516,564,636,660,708,

%T 732,780,804,852,876,924,948,996,1020,1068,1092,1140,1164,1212,1236,

%U 1284,1308,1356,1380,1428,1524,1540,1572,1596,1644,1668,1716,1740,1788,1812,1820

%N Nonsquarefree numbers not in A225353; equivalently, nonsquarefree numbers in A225354.

%C 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 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 If k is a term, then so is m * k where m is squarefree and coprime to k. - _Robert Israel_, Apr 21 2024

%C Comparison with other similar sequences:

%C For values up to and including a(2000)=76044:

%C b(n): | 12*A276378| 12*A007310| 12*A038179| 4*A243128| A357686

%C --------------------------------------------------------------------------------

%C # a(n) not in b(n) | 73| 70| 74| 0| 1

%C # b(n) not in a(n) | 0| 186| 188| 69| 69

%C First a(n) not in b(n)| a(40)=1540| a(40)=1540| a(1)=12| - | a(1)=12

%C First b(n) not in a(n)| - | 12*b(9)=300| 12*b(1)=24| 4*b(5)=140| b(4)=140

%F Equals A013929\A225353 and also A225354\A005117.

%e 12 is a term since 12 = 2^2*3 and 12 = 1 + 2 + 3 + 6.

%p filter:= proc(n) local P,z,d;

%p if numtheory:-issqrfree(n) then return false fi;

%p P:= mul(1+z^d, d = select(numtheory:-issqrfree,numtheory:-divisors(n)));

%p coeff(P,z,n) > 0

%p end proc:

%p select(filter, [$1..2000]); # _Robert Israel_, Apr 21 2024

%t filter[n_] := Module[{P, z, d},

%t If[SquareFreeQ[n], Return[False]];

%t P = Product[1 + z^d, {d, Select[Divisors[n], SquareFreeQ]}];

%t Coefficient[P, z, n] > 0];

%t Select[Range[2000], If[filter[#], Print[#]; True, False]&] (* _Jean-François Alcover_, May 28 2024, after _Robert Israel_ *)

%Y Cf. A005117 (squarefree numbers), A013929 (nonsquarefree numbers), A225353, A225354, A007310, A038179, A243128, A276378, A357686.

%K nonn

%O 1,1

%A _Miles Englezou_, Apr 20 2024