A129575 -id:A129575 - cp's OEIS frontend

A336254 Exponential barely abundant numbers: exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller exponential abundant number.

900, 1764, 3600, 4356, 4500, 4900, 12348, 47916, 79092, 112500, 605052, 2812500, 13366548, 29647548, 89139564, 231708348, 701538156, 1757812500, 14772192228, 32179382604, 43945312500, 71183762748, 620995547124, 990454107996, 3417547576788, 3488004374652, 10271220141996

Offset: 1

Amiram Eldar, Jul 14 2020

nonn

The exponential abundancy of a number k is esigma(k)/k, where esigma is the sum of exponential divisors of k (A051377).

All the terms are powerful numbers (A001694) because esigma(k)/k depends only on the powerful part of k (A057521). - Amiram Eldar, May 06 2025

			The first 6 exponential abundant numbers, 900, 1764, 3600, 4356, 4500 and 4900, have decreasing values of exponential abundancy: 2.4, 2.285..., 2.2, 2.181..., 2.08, 2.057... and therefore they are in this sequence. The next exponential abundant number with a lower exponential abundancy is 12348 with eisgma(12348)/12348 = 2.040...

Amiram Eldar, Table of n, a(n) for n = 1..36

The exponential version of A071927.
Subsequence of A001694 and A328136.
Cf. A051377, A057521, A129575, A336253.
Similar sequences: A188263, A302570, A302571, A335054.

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; rm = 3; s={}; Do[r = esigma[n]/n; If[r <= 2, Continue[]]; If[r < rm, rm = r; AppendTo[s, n]], {n, 1, 10^6}]; s

a(23)-a(27) from Amiram Eldar, May 06 2025

A357685 Numbers k such that A293228(k) > k.

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 102, 114, 132, 138, 140, 156, 174, 186, 204, 210, 222, 228, 246, 258, 276, 282, 318, 330, 348, 354, 366, 372, 390, 402, 420, 426, 438, 444, 462, 474, 492, 498, 510, 516, 534, 546, 564, 570, 582, 606, 618, 636, 642, 654, 660, 678, 690

Offset: 1

Amiram Eldar, Oct 09 2022

nonn

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... .

			30 is a term since its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15} and their sum is 42 > 30.
60 is a term since its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15, 30} and their sum is 72 > 60.

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

Cf. A005117, A293228.
Disjoint union of A087248 and A357686.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

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

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};

A383695 Exponential infinitary abundant numbers that are not exponential unitary abundant: numbers k such that A361175(k) > 2*k >= A322857(k).

Original entry on oeis.org

476985600, 815673600, 1018886400, 1177862400, 1493049600, 2014214400, 2373638400, 2712326400, 3756614400, 3863865600, 4744454400, 5218617600, 5246841600, 6234681600, 7928121600, 8108755200, 8245036800, 8972409600, 9062726400, 9824774400, 10502150400, 10603756800

Offset: 1

Amiram Eldar, May 05 2025

nonn

Exponential infinitary abundant numbers are numbers k such that A361175(k) > 2*k.
All the exponential unitary abundant numbers (A383693) are also exponential infinitary abundant numbers. There are numbers that are exponential infinitary abundant and not exponential unitary abundant. The least is: a(1) = 476985600, which is the 427970th exponential infinitary abundant number.
All the terms are nonsquarefree numbers (A013929), since A361175(k) = k if k is a squarefree number (A005117).
The asymptotic density of this sequence is Sum_{n>=1} f(A383696(n)) = 1.9875...*10^(-9), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). The relative density of this sequence within the exponential infinitary abundant numbers is 2.215... * 10^(-6).

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

Cf. A005117, A322857, A361175.
Subsequence of A013929 and A129575.
A383696 is a subsequence.

seq[max_] := Module[{prim = seqA383696[max], s = {}, sq}, Do[sq = Select[Range[Floor[max/p]], CoprimeQ[p, #] && SquareFreeQ[#] &]; s = Join[s, p*sq], {p, prim}]; Union[s]]; seq[10^10] (* using the function seqA383696 from A383696 *)

list(lim) = {my(p = listA383696(lim), s = []); for(i = 1, #p, s = concat(s, apply(x -> p[i]*x, select(x -> gcd(x, p[i]) == 1 && issquarefree(x), vector(lim\p[i], j, j))))); Set(s);} \\ using the function listA383696 from A383696

A336680 Exponential admirable numbers: numbers k such that there is a proper exponential divisor d of k such that esigma(k) - 2d = 2k, where esigma is the sum of exponential divisors function (A051377).

Original entry on oeis.org

900, 1764, 4356, 4500, 4900, 6084, 6300, 7056, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 35280, 36900, 38700, 40572, 42300, 42588, 47700

Offset: 1

Amiram Eldar, Jul 30 2020

nonn

Equivalently, numbers that are equal to the sum of their proper exponential divisors, with one of them taken with a minus sign.

			900 is a term since 900 = 30 + 60 + 90 + 150 - 180 + 300 + 450 is the sum of its proper exponential divisors with one of them, 180, taken with a minus sign.

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

The exponential version of A111592.
Subsequence of A129575.
Similar sequences: A328328, A334972, A334974.
Cf. A051377, A322791.

dQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m]); expDivQ[n_, d_] := Module[{ft = FactorInteger[n]}, And @@ MapThread[dQ, {ft[[;; , 2]], IntegerExponent[d, ft[[;; , 1]]]}]]; esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; expAdmQ[n_] := (ab = esigma[n] - 2*n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && expDivQ[n, ab/2]; Select[Range[50000], expAdmQ]

A360525 Numbers k such that A360522(k) > 2*k.

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 140, 150, 156, 168, 174, 180, 186, 204, 210, 222, 228, 246, 252, 258, 276, 282, 294, 300, 318, 330, 348, 354, 360, 366, 372, 390, 402, 420, 426, 438, 444, 462, 474, 492, 498, 510, 516, 534, 546, 564

Offset: 1

Amiram Eldar, Feb 10 2023

nonn

First differs from A308127 at n = 15.
Analogous to abundant numbers (A005101) with A360522 instead of A000203.
Subsequence of A005101 because A360522(n) <= A000203(n) for all n.
The least odd term is a(1698) = A360526(1) = 15015.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 8, 95, 1135, 10890, 110867, 1104596, 11048123, 110534517, 1105167384, 11051009278, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1105...

			30 is a term since A360522(30) = 72 > 2*30.

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

Cf. A000203, A308127, A360522, A360524, A360526.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

f[p_, e_] := p^e + e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := s[n] > 2*n; Select[Range[1000], q]

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

A348627 Numbers that are both exponential and nonexponential abundant numbers.

Original entry on oeis.org

3600, 4500, 6300, 7056, 8100, 8820, 9900, 14700, 21780, 22500, 25200, 30420, 31500, 35280, 39600, 46800, 49500, 52020, 56700, 58500, 61200, 61740, 64980, 68400, 69300, 76500, 77616, 81900, 82800, 85500, 88200, 89100, 91728, 95220, 97020, 103500, 104400, 105300

Offset: 1

Amiram Eldar, Oct 26 2021

nonn

			3600 is a term since A051377(3600) = 7920 > 2*3600 and A160135(3600) = 4573 > 3600.

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

Cf. A051377, A160135.
Intersection of A129575 and A348604.
Subsequence of A068403.
Similar sequence: A348523.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; Select[Range[10^5], (e = esigma[#]) > 2*# && DivisorSigma[1, #] - e > # &]

cp's OEIS Frontend

A336254 Exponential barely abundant numbers: exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller exponential abundant number.

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A357685 Numbers k such that A293228(k) > k.

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A383695 Exponential infinitary abundant numbers that are not exponential unitary abundant: numbers k such that A361175(k) > 2*k >= A322857(k).

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A336680 Exponential admirable numbers: numbers k such that there is a proper exponential divisor d of k such that esigma(k) - 2*d = 2*k, where esigma is the sum of exponential divisors function (A051377).

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A360525 Numbers k such that A360522(k) > 2*k.

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A348627 Numbers that are both exponential and nonexponential abundant numbers.

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A336680 Exponential admirable numbers: numbers k such that there is a proper exponential divisor d of k such that esigma(k) - 2d = 2k, where esigma is the sum of exponential divisors function (A051377).