A348604 -id:A348604 - cp's OEIS frontend

A348631 Nonexponential weird numbers: nonexponential abundant numbers (A348604) that are not equal to the sum of any subset of their nonexponential divisors.

70, 4030, 5830, 10430, 10570, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810, 20510, 21490, 21770, 21910, 22190, 23170, 23590, 24290

Offset: 1

Amiram Eldar, Oct 26 2021

nonn

			70 is a term since the sum of its nonexponential divisors, {1, 2, 5, 7, 10, 14, 35}, is 74 > 70, and no subset of these divisors sums to 70.

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

Cf. A160135, A348604.

Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948, A348525.

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]]]}]]; neDivs[1] = {}; neDivs[n_] := Module[{d = Divisors[n]}, Select[d, ! expDivQ[n, #] &]]; nesigma[n_] := Total@neDivs[n]; neAbundantQ[n_] := nesigma[n] > n; neWeirdQ[n_] := neAbundantQ[n] && Module[{d = neDivs[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[6000], neWeirdQ]

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

Original entry on oeis.org

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;}

A357605 Numbers k such that A162296(k) > 2*k.

Original entry on oeis.org

36, 48, 72, 80, 96, 108, 120, 144, 160, 162, 168, 180, 192, 200, 216, 224, 240, 252, 264, 270, 280, 288, 300, 312, 320, 324, 336, 352, 360, 378, 384, 392, 396, 400, 408, 416, 432, 448, 450, 456, 468, 480, 486, 500, 504, 528, 540, 552, 560, 576, 588, 594, 600, 612

Offset: 1

Amiram Eldar, Oct 06 2022

nonn

The least odd term is a(470) = A357607(1) = 4725.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 5, 92, 1011, 10160, 102125, 1022881, 10231151, 102249758, 1022781199, 10229781638, ... . Apparently, the asymptotic density of this sequence exists and equals 0.102... .
An analog of abundant numbers, in which the divisor sum is restricted to nonsquarefree divisors. - Peter Munn, Oct 26 2022

			36 is a term since A162296(36) = 79 > 2*36.

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

Cf. A162296.
Subsequence of A005101 and A013929.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; Select[Range[2, 1000], q]

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;}

A348605 Odd nonexponential abundant numbers: odd numbers k such that A160135(k) > k.

Original entry on oeis.org

8505, 10395, 12285, 15015, 16065, 17955, 19635, 21735, 21945, 23205, 25515, 25935, 26565, 28875, 31185, 31395, 33495, 33915, 34125, 35805, 36855, 39585, 41055, 42315, 42735, 45885, 47355, 48195, 49665, 50505, 51765, 53865, 54285, 55965, 56595, 58695, 61215, 64155

Offset: 1

Amiram Eldar, Oct 25 2021

nonn

The odd terms of A348604.

The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 1, 51, 360, 4117, 39803, 418663, 4099004, ... Apparently this sequence has an asymptotic density 0.0004...

			8505 is a term since A160135(8505) = 8862 > 8505.

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

Cf. A160135.
Subsequence of A005231 and A348604.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A348275.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[1, 65000, 2], q]

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};

A348629 Nonexponential highly abundant numbers: numbers m such that nesigma(m) > nesigma(k) for all k < m, where nesigma(k) is the sum of nonexponential divisors of n (A160135).

Original entry on oeis.org

1, 6, 10, 12, 18, 24, 30, 42, 48, 54, 60, 78, 84, 90, 96, 120, 168, 192, 210, 240, 270, 312, 330, 360, 384, 420, 480, 630, 672, 840, 960, 1056, 1080, 1248, 1320, 1440, 1560, 1680, 1890, 1920, 2280, 2310, 2400, 2520, 2640, 2688, 3000, 3120, 3240, 3360, 4200, 4320

Offset: 1

Amiram Eldar, Oct 26 2021

nonn

The corresponding record values are 1, 6, 8, 10, 15, 30, 42, 54, 58, 60, 78, ... (see the link for more values).

			The first 6 values of nesigma(k), for k = 1 to 6 are 1, 1, 1, 1, 1 and 6. The record values, 1 and 6, occur at 1 and 6, the first 2 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..577
Amiram Eldar, Table of n, a(n), A160135(a(n)) for n = 1..577

Cf. A160135, A348604.
The nonexponential version of A002093.
Similar sequences: A285614, A292983, A327634, A328134, A329883, A348272.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[1] = 1; s[n_] := DivisorSigma[1, n] - esigma[n]; seq = {}; sm = -1; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

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;}

A348606 Numbers k such that k and k+1 are both nonexponential abundant numbers.

Original entry on oeis.org

21735, 76544, 170624, 301664, 345344, 348704, 382304, 739935, 862784, 1218944, 1262624, 1272704, 1314495, 1370655, 1376864, 1539615, 1558304, 1707615, 1718144, 1761375, 1845375, 1890944, 1926015, 2100735, 2132864, 2223584, 2415104, 2463615, 2581215, 2675295, 2747744

Offset: 1

Amiram Eldar, Oct 25 2021

nonn

			21735 is a term since A160135(21735) = 21930 > 21735 and A160135(21736) = 23230 > 21736.

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

Cf. A160135.
Subsequence of A096399 and A348604.
Similar sequences: A318167, A327635, A327942, A331412, A348276.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[1, 3*10^6], q[#] && q[#+1] &]

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

A348631 Nonexponential weird numbers: nonexponential abundant numbers (A348604) that are not equal to the sum of any subset of their nonexponential divisors.

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

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A357605 Numbers k such that A162296(k) > 2*k.

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

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A348605 Odd nonexponential abundant numbers: odd numbers k such that A160135(k) > k.

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

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A348629 Nonexponential highly abundant numbers: numbers m such that nesigma(m) > nesigma(k) for all k < m, where nesigma(k) is the sum of nonexponential divisors of n (A160135).

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

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