A379029 -id:A379029 - cp's OEIS frontend

A380929 Numbers k such that A380845(k) > 2*k.

36, 72, 84, 140, 144, 168, 180, 264, 270, 280, 288, 300, 336, 360, 372, 392, 450, 520, 528, 532, 540, 558, 560, 576, 594, 600, 612, 620, 672, 720, 744, 756, 780, 784, 840, 900, 930, 1036, 1040, 1050, 1056, 1064, 1068, 1080, 1092, 1116, 1120, 1134, 1152, 1170, 1180, 1188, 1200

Offset: 1

Amiram Eldar, Feb 08 2025

Analogous to abundant numbers (A005101) with A380845 instead of A000203.

			36 is a term since A380845(36) = 84 > 2 * 36 = 72.

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

Cf. A000203, A380845, A380846.
Subsequence of A005101.
Subsequences: A380847, A380848, A380930, A380931.
Similar sequences: A034683, A064597, A087248, A129575, A129656, A292982, A348274, A348604, A379029.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1200], q]

isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}

A379031 Odd modified exponential abundant numbers: odd numbers k such that A241405(k) > 2*k.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035, 116655

Offset: 1

Amiram Eldar, Dec 14 2024

First differs from its subsequences A112643 and A249263 at n = 51: a(51) = 195195 is not a term of these two sequences.
First differs from its subsequence A129485 at n = 363: a(363) = 2537535 is not a term of A129485.
First differs from A339938 at n = 28: A339938(28) = 75075 is not a term of this sequence.
First differs from A360526 at n = 46: A360526(46) = 165165 is not a term of this sequence.

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

Intersection of A005408 and A379029.
Subsequence of A005231.
Subsequences: A112643, A129485, A249263.
Cf. A241405.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A339938, A348275, A360526.

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[1, 10^5, 2], meAbQ]

is(k) = if(!(k%2), 0, my(f=factor(k)); prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))) > 2*k);

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

Original entry on oeis.org

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

Amiram Eldar, Dec 14 2024

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... .

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

Intersection of A013929 and A379029.
Complement of A087248 within A379029.
Cf. A005117, A241405.

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]

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);}

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A380929 Numbers k such that A380845(k) > 2*k.

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A379031 Odd modified exponential abundant numbers: odd numbers k such that A241405(k) > 2*k.

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

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