A225353 -id:A225353 - cp's OEIS frontend

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

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.

A225245 Number of partitions of n into distinct squarefree divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 4, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 3, 1, 1, 0, 0, 1, 3, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 1

Offset: 0

Reinhard Zumkeller, May 05 2013

nonn

a(n) <= A033630(n);
a(n) = A033630(n) iff n is squarefree: a(A005117(n)) = A033630(A005117(n));
a(A225353(n)) = 0; a(A225354(n)) > 0.

			a(2*3)     = a(6)  = #{6, 3+2+1} = 2;
a(2*2*3)   = a(12) = #{6+3+2+1} = 1;
a(2*3*5)   = a(30) = #{30, 15+10+5, 15+10+3+2, 15+6+5+3+1} = 4;
a(2*2*3*5) = a(60) = #{30+15+10+5, 30+15+10+3+2, 30+15+6+5+3+1} = 3;
a(2*3*7)   = a(42) = #{42, 21+14+7, 21+14+6+1} = 3;
a(2*2*3*7) = a(84) = #{42+21+14+7, 42+21+14+6+1} = 2.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (5000 terms from Reinhard Zumkeller)
Noah Lebowitz-Lockard and Joseph Vandehey, On the number of partitions of a number into distinct divisors, arXiv:2402.08119 [math.NT], 2024. See p. 2.

Cf. A005117, A008683, A033630, A206778, A008966, A225244, A087188, A225353, A225354.

a225245 n = p (a206778_row n) n where
   p _      0 = 1
   p []     _ = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

a[n_] := If[n == 0, 1, Coefficient[Product[If[MoebiusMu[d] != 0, 1+x^d, 1], {d, Divisors[n]}], x, n]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 08 2021, after Ilya Gutkovskiy *)

a(n) = [x^n] Product_{d|n, mu(d) != 0} (1 + x^d), where mu() is the Moebius function (A008683). - Ilya Gutkovskiy, Jul 26 2017

A225354 Numbers that can be written as sum of distinct squarefree divisors.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Reinhard Zumkeller, May 05 2013

nonn

A225245(a(n)) > 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A225353 (complement).

import Data.List (findIndices)
a225354 n = a225354_list !! (n-1)
a225354_list = map (+ 1) $ findIndices (> 0) a225245_list

f[n_] := f[n] = Coefficient[Product[If[MoebiusMu[d] != 0, 1 + x^d, 1], {d, Divisors[n]}], x, n];
ParallelTable[If[f[k] > 0, k, Nothing], {k, 1, 1000}] (* Jean-François Alcover, May 04 2024, after Ilya Gutkovskiy in A225245 *)

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

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A225245 Number of partitions of n into distinct squarefree divisors of n.

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A225354 Numbers that can be written as sum of distinct squarefree divisors.

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