A225354 -id:A225354 - 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

A225353 Numbers having no partition into distinct squarefree divisors.

Original entry on oeis.org

4, 8, 9, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 63, 64, 68, 72, 75, 76, 80, 81, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 126, 128, 135, 136, 140, 144, 147, 148, 150, 152, 153, 160, 162, 164

Offset: 1

Reinhard Zumkeller, May 05 2013

nonn

A225245(a(n)) = 0.
Verified up to a(366) = 1000, a(n) is also the order of a finite group G for which |Out(G)|>|G| for at least one group of order a(n), Out(G) being the outer automorphism group of G. - Miles Englezou, Apr 19 2024
By definition, a(n) is nonsquarefree for every n, since every squarefree number m has a trivial partition into distinct squarefree divisors m = m. - Miles Englezou, Apr 20 2024
If k is a term then so is k*m where m|k. - David A. Corneth, Apr 27 2024

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

Cf. A013929, A225245, A225354 (complement).

import Data.List (elemIndices)
a225353 n = a225353_list !! (n-1)
a225353_list = map (+ 1) $ elemIndices 0 a225245_list

f[n_] := Coefficient[Product[If[MoebiusMu[d] != 0, 1 + x^d, 1], {d, Divisors[n]}], x, n];
Select[Range[200], f[#] == 0&] (* Jean-François Alcover, May 04 2024, after Ilya Gutkovskiy in A225245 *)

A349019 Modified e-perfect numbers: numbers k such that A348963(k) | k.

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, 36, 37, 38, 39, 40, 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

Offset: 1

Amiram Eldar, Nov 06 2021

nonn

First differs from A225354 at n = 25.
Not to be confused with modified exponential perfect numbers (A323757).
Sándor (2006) showed that the exponential harmonic numbers of type 2 (A348964) are terms in this sequence.
All the squarefree numbers are terms (A005117), since A348963(k) = 1 if k is squarefree.

			12 is a term since A348963(12) = 3 is a divisor of 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

A005117, A348964 and A349020 are subsequences.

Cf. A225354, A323757, A348963.

f[p_, e_] := p^e/DivisorSum[e, p^(e - #) &]; modEPerfQ[1] = True; modEPerfQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], modEPerfQ]

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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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A225353 Numbers having no partition into distinct squarefree divisors.

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A349019 Modified e-perfect numbers: numbers k such that A348963(k) | k.

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