A357685 - OEIS

%I #10 Oct 12 2022 04:34:49

%S 30,42,60,66,70,78,84,102,114,132,138,140,156,174,186,204,210,222,228,

%T 246,258,276,282,318,330,348,354,366,372,390,402,420,426,438,444,462,

%U 474,492,498,510,516,534,546,564,570,582,606,618,636,642,654,660,678,690

%N Numbers k such that A293228(k) > k.

%C 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... .

%H Amiram Eldar, <a href="/A357685/b357685.txt">Table of n, a(n) for n = 1..10000</a>

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

%e 60 is a term since its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15, 30} and their sum is 72 > 60.

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

%o (PARI) 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};

%Y Cf. A005117, A293228.

%Y Disjoint union of A087248 and A357686.

%Y Subsequence of A005101.

%Y Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

%K nonn

%O 1,1

%A _Amiram Eldar_, Oct 09 2022