A379030 Nonsquarefree modified exponential abundant numbers: nonsquarefree numbers k such that A241405(k) > 2*k.

120, 150, 168, 270, 294, 420, 630, 660, 726, 750, 780, 840, 924, 990, 1014, 1020, 1050, 1092, 1140, 1170, 1320, 1380, 1386, 1428, 1470, 1530, 1560, 1596, 1638, 1650, 1710, 1734, 1740, 1848, 1860, 1890, 1950, 2040, 2058, 2070, 2142, 2166, 2184, 2220, 2280, 2394

Offset: 1

Author

Amiram Eldar, Dec 14 2024

Keywords

• nonn
• easy

Comments

All the squarefree abundant numbers (A087248) are also modified exponential abundant numbers (A379029). This sequence lists the terms of A379029 that are not in A087248.

The numbers of terms that do no exceed 10^k, for k = 3, 4, ..., are 14, 211, 2090, 21236, 212744, 2123071, 21235175, 212450318, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0212... .

Links

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

Crossrefs

Intersection of A013929 and A379029.

Complement of A087248 within A379029.

Cf. A005117, A241405.

Programs

Mathematica

f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := Times @@ f @@@ FactorInteger[n]; nsmeAbQ[n_] := !SquareFreeQ[n] && mesigma[n] > 2*n; Select[Range[2400], nsmeAbQ]

PARI

is(n) = {my(f=factor(n)); if(issquarefree(f), 0, prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))) > 2*n);}