A357686 -id:A357686 - cp's OEIS frontend

A357685 Numbers k such that A293228(k) > k.

30, 42, 60, 66, 70, 78, 84, 102, 114, 132, 138, 140, 156, 174, 186, 204, 210, 222, 228, 246, 258, 276, 282, 318, 330, 348, 354, 366, 372, 390, 402, 420, 426, 438, 444, 462, 474, 492, 498, 510, 516, 534, 546, 564, 570, 582, 606, 618, 636, 642, 654, 660, 678, 690

Offset: 1

Amiram Eldar, Oct 09 2022

nonn

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 7, 79, 843, 8230, 83005, 826875, 8275895, 82790525, 827718858, 8276571394, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0827... .

			30 is a term since its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15} and their sum is 42 > 30.
60 is a term since 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

Cf. A005117, A293228.
Disjoint union of A087248 and A357686.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

s[n_] := Times @@ (1 + (f = FactorInteger[n])[[;; , 1]]) - If[AllTrue[f[[;;, 2]], # == 1 &], n, 0]; Select[Range[2, 1000], s[#] > # &]

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

A357687 Nonsquarefree numbers k such that A048250(k) > 2*k.

Original entry on oeis.org

401120980260, 14841476269620, 16445960190660, 17248202151180, 18852686072220, 608500527054420, 638183479593660, 697549384672140, 707176288198380, 772960128961020, 810665501105460, 26165522663340060, 28599524771557740, 29994623540902020, 33237285545323860, 1229779565176982820

Offset: 1

Amiram Eldar, Oct 09 2022

nonn

The squarefree numbers k such that A048250(k) > 2*k are the squarefree abundant numbers (A087248).

The least odd term is 3*prime(553)#/2 = 3.735...*10^1709.

Subsequence of A005101 and A013929.

Cf. A005117, A048250, A087248, A357685, A357686.

q[n_] := Module[{f = FactorInteger[n]}, AnyTrue[f[[;;, 2]], # > 1 &] && Times @@ (f[[;;, 1]]+1) > 2*n];

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

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A357685 Numbers k such that A293228(k) > k.

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A357687 Nonsquarefree numbers k such that A048250(k) > 2*k.

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

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