A243128 -id:A243128 - cp's OEIS frontend

A357686 Nonsquarefree numbers k such that A293228(k) > k.

60, 84, 132, 140, 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

Amiram Eldar, Oct 09 2022

nonn

The squarefree numbers k such that A293228(k) > k are the squarefree abundant numbers (A087248).
If k > 3 is a term of A243128 then 4*k is a term.
The least odd term is (3/2)*prime(17)# = 2884140525231318958605.
The least term that is coprime to 6 is (5/6)*prime(1245)# = 5.629...*10^4361.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 2, 26, 287, 2725, 27660, 275298, 2754638, 27556849, 275538900, 2755151247, ... . Apparently, the asymptotic density of this sequence exists and equals 0.02755... .

			60 = 2^2 * 15 is a term since it is nonsquarefree, its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15, 30} and their sum is 72 > 60.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A013929 and A357685.
Subsequence of A005101.
Cf. A087248, A243128.

q[n_] := AnyTrue[(f = FactorInteger[n])[[;;, 2]], # > 1 &] && Times @@ (1 + f[[;; , 1]]) > n; Select[Range[2, 2000], q]

is(n) = {my(f = factor(n)); if(n == 1 || vecmax(f[,2]) == 1, return(0)); prod(i=1, #f~, f[i,1]+1) > n};

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

Original entry on oeis.org

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

Miles Englezou, Apr 20 2024

nonn

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

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

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

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

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 *)

Equals A013929\A225353 and also A225354\A005117.

cp's OEIS Frontend

A357686 Nonsquarefree numbers k such that A293228(k) > k.

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