A129656 -id:A129656 - cp's OEIS frontend

A335054 Infinitary barely abundant numbers: infinitary abundant numbers whose infinitary abundancy is closer to 2 than that of any smaller infinitary abundant number.

24, 30, 40, 54, 56, 70, 88, 104, 642, 654, 678, 726, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1014, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1398, 1434, 1446, 1506, 1536, 1542, 1578, 1596, 2406, 2454, 2514, 2526, 2586, 2598, 2634

Offset: 1

Amiram Eldar, May 21 2020

nonn

The infinitary abundancy of a number k is isigma(k)/k, where isigma(k) is the sum of infinitary divisors of k (A049417).

			The infinitary abundancies of the first terms are 2.5, 2.4, 2.25, 2.222..., 2.142..., 2.057..., ...

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

The infinitary version of A071927.

Cf. A049417, A129656, A302570, A302571.

fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; seq = {}; r = 3; Do[s = isigma[n]/n; If[s > 2 && s < r, AppendTo[seq, n]; r = s], {n, 1, 3000}]; seq

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

Original entry on oeis.org

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

Amiram Eldar, Dec 14 2024

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... .

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

Cf. A000203, A241405, A323757, A323758, A323759, A379027.
Subsequence of A005101.
Subsequences: A034683, A087248, A379030, A379031.
Similar sequences: A064597, A129575, A129656, A292982, A348274, A348604.

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]

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

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};

A348523 Numbers that are both infinitary and noninfinitary abundant numbers.

Original entry on oeis.org

960, 1440, 1800, 2016, 2400, 2940, 3240, 3528, 3780, 4536, 4860, 6720, 7260, 8640, 10080, 10140, 10560, 12096, 12480, 12600, 13860, 14784, 15120, 15360, 15840, 16320, 16380, 16800, 17472, 17640, 18240, 18480, 18720, 18900, 19008, 19800, 20160, 21420, 21600, 21840

Offset: 1

Amiram Eldar, Oct 21 2021

nonn

Apparently, the smallest odd term is 9170790153525.

			960 is a term since A049417(960) = 2040 > 2*960 = 1920 and A348271(960) = 1008 > 960.

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

Intersection of A129656 and A348274.
Subsequence of A068403.
Cf. A049417, A348271.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := (i = isigma[n]) > 2*n && DivisorSigma[1, n] - i > n; Select[Range[10^4], q]

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;}

A374788 Numbers whose infinitary divisors have a mean infinitary abundancy index that is larger than 2.

Original entry on oeis.org

7560, 9240, 10920, 83160, 98280, 120120, 120960, 128520, 143640, 147840, 154440, 157080, 173880, 174720, 175560, 185640, 189000, 190080, 201960, 207480, 212520, 216216, 219240, 224640, 225720, 228480, 231000, 234360, 238680, 251160, 255360, 266112, 266760, 267960

Offset: 1

Amiram Eldar, Jul 20 2024

nonn

Numbers k such that A374786(k)/A374787(k) > 2.

The least odd term is 17737266779965459404793703604641625, and the least term that is coprime to 6 is 5^7 * (7 * 11 * ... * 23)^3 * 29 * 31 * ... * 751 = 3.140513... * 10^329.

			7560 is a term since A374786(7560)/A374787(7560) = 1045/512 = 2.041... > 2.

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

Cf. A129656, A374786, A374787.

Similar sequences: A245214, A374785.

f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)])); q[1] = False; q[n] := Times @@ (1 + 1/(2*Flatten@ (f @@@ FactorInteger[n]))) > 2; Select[Range[300000], q]

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

A372300 Numbers k such that k and k+1 are both primitive infinitary abundant numbers (definition 1, A372298).

Original entry on oeis.org

812889, 3181815, 20787584, 181480695, 183872535, 307510664, 337206344, 350158808, 523403264, 744074624, 868421504, 1063361144, 1955365125, 2076191864, 2578966215, 3672231255, 4185590408, 5032685384, 7158001304, 8348108535, 10784978295, 16264812135, 20917209495, 24514454055

Offset: 1

Amiram Eldar, Apr 25 2024

nonn

The corresponding sequence with definition 2 (A372299) coincides with this sequence for the first 24 terms.

Subsequence of A129656, A327635 and A372298.
Cf. A372299.
Similar sequences: A283418, A330872, A361935.

isidiv(d, f) = {my(bne,bde); if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
isab(n) = isigma(n) > 2*n;
isprim(n) = select(x -> x= 2*x, idivs(n)) == [];
lista(kmax) = {my(is1 = 0, is2); for(k = 2, kmax, is2 = isab(k); if(is1 && is2, if(isprim(k-1) && isprim(k), print1(k-1, ", "))); is1 = is2);}

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A335054 Infinitary barely abundant numbers: infinitary abundant numbers whose infinitary abundancy is closer to 2 than that of any smaller infinitary abundant number.

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

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

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A348523 Numbers that are both infinitary and noninfinitary abundant numbers.

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

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A374788 Numbers whose infinitary divisors have a mean infinitary abundancy index that is larger than 2.

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A372300 Numbers k such that k and k+1 are both primitive infinitary abundant numbers (definition 1, A372298).

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