login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Nonsquarefree numbers k such that A293228(k) > k.
4

%I #9 Oct 12 2022 04:31:47

%S 60,84,132,140,156,204,228,276,348,372,420,444,492,516,564,636,660,

%T 708,732,780,804,852,876,924,948,996,1020,1068,1092,1140,1164,1212,

%U 1236,1284,1308,1356,1380,1428,1524,1540,1572,1596,1644,1668,1716,1740,1788,1812,1820

%N Nonsquarefree numbers k such that A293228(k) > k.

%C The squarefree numbers k such that A293228(k) > k are the squarefree abundant numbers (A087248).

%C If k > 3 is a term of A243128 then 4*k is a term.

%C The least odd term is (3/2)*prime(17)# = 2884140525231318958605.

%C The least term that is coprime to 6 is (5/6)*prime(1245)# = 5.629...*10^4361.

%C 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... .

%H Amiram Eldar, <a href="/A357686/b357686.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

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

%o (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};

%Y Intersection of A013929 and A357685.

%Y Subsequence of A005101.

%Y Cf. A087248, A243128.

%K nonn

%O 1,1

%A _Amiram Eldar_, Oct 09 2022