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

12, 60, 84, 132, 156, 204, 228, 276, 348, 372, 420, 444, 492, 516, 564, 636, 660, 708, 732, 780, 804, 852, 876, 924, 948, 996, 1020, 1068, 1092, 1140, 1164, 1212, 1236, 1284, 1308, 1356, 1380, 1428, 1524, 1540, 1572, 1596, 1644, 1668, 1716, 1740, 1788, 1812, 1820

Offset: 1

Author

Miles Englezou, Apr 20 2024

Keywords

• nonn

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:

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

--------------------------------------------------------------------------------

# 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

Examples

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

Crossrefs

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

Programs

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 *)

Formula

Equals A013929\A225353 and also A225354\A005117.