A379029 Modified exponential abundant numbers: numbers k such that A241405(k) > 2*k.

30, 42, 66, 70, 78, 102, 114, 120, 138, 150, 168, 174, 186, 210, 222, 246, 258, 270, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822

Offset: 1

Author

Amiram Eldar, Dec 14 2024

Keywords

• nonn
• easy

Comments

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

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

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... .

Links

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

Crossrefs

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

Subsequence of A005101.

Subsequences: A034683, A087248, A379030, A379031.

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

Programs

Mathematica

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]

PARI

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