A379029 - OEIS

%I #9 Dec 16 2024 02:17:21

%S 30,42,66,70,78,102,114,120,138,150,168,174,186,210,222,246,258,270,

%T 282,294,318,330,354,366,390,402,420,426,438,462,474,498,510,534,546,

%U 570,582,606,618,630,642,654,660,678,690,714,726,750,762,770,780,786,798,822

%N Modified exponential abundant numbers: numbers k such that A241405(k) > 2*k.

%C All the squarefree abundant numbers (A087248) are terms since A241405(k) = A000203(k) for a squarefree number k.

%C If k is a term and m is coprime to k them k*m is also a term.

%C The numbers of terms that do no exceed 10^k, for k = 2, 3, ..., are 5, 67, 767, 7595, 76581, 764321, 7644328, 76468851, 764630276, ... . Apparently, the asymptotic density of this sequence exists and equals 0.07646... .

%H Amiram Eldar, <a href="/A379029/b379029.txt">Table of n, a(n) for n = 1..10000</a>

%t f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := Times @@ f @@@ FactorInteger[n]; meAbQ[n_] := mesigma[n] > 2*n; Select[Range[1000], meAbQ]

%o (PARI) is(n) = {my(f=factor(n)); prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))) > 2*n;}

%Y Cf. A000203, A241405, A323757, A323758, A323759, A379027.

%Y Subsequence of A005101.

%Y Subsequences: A034683, A087248, A379030, A379031.

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

%K nonn,easy,new

%O 1,1

%A _Amiram Eldar_, Dec 14 2024