OFFSET
1,1
COMMENTS
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.
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.
If k is a term, then so is m * k where m is squarefree and coprime to k. - Robert Israel, Apr 21 2024
Comparison with other similar sequences:
For values up to and including a(2000)=76044:
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# a(n) not in b(n) | 73| 70| 74| 0| 1
# b(n) not in a(n) | 0| 186| 188| 69| 69
First a(n) not in b(n)| a(40)=1540| a(40)=1540| a(1)=12| - | a(1)=12
First b(n) not in a(n)| - | 12*b(9)=300| 12*b(1)=24| 4*b(5)=140| b(4)=140
EXAMPLE
12 is a term since 12 = 2^2*3 and 12 = 1 + 2 + 3 + 6.
MAPLE
filter:= proc(n) local P, z, d;
if numtheory:-issqrfree(n) then return false fi;
P:= mul(1+z^d, d = select(numtheory:-issqrfree, numtheory:-divisors(n)));
coeff(P, z, n) > 0
end proc:
select(filter, [$1..2000]); # Robert Israel, Apr 21 2024
MATHEMATICA
filter[n_] := Module[{P, z, d},
If[SquareFreeQ[n], Return[False]];
P = Product[1 + z^d, {d, Select[Divisors[n], SquareFreeQ]}];
Coefficient[P, z, n] > 0];
Select[Range[2000], If[filter[#], Print[#]; True, False]&] (* Jean-François Alcover, May 28 2024, after Robert Israel *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Miles Englezou, Apr 20 2024
STATUS
approved