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

Author

Amiram Eldar, Oct 09 2022

Keywords

• nonn

Comments

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... .

Examples

			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.
		

Links

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

Crossrefs

Intersection of A013929 and A357685.

Subsequence of A005101.

Cf. A087248, A243128.

Programs

Mathematica

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

PARI

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