A094889 -id:A094889 - cp's OEIS frontend

A327942 Numbers k such that both k and k+1 are nonunitary abundant numbers (A064597).

165375, 893024, 1047375, 1576575, 2282175, 2304224, 2858624, 3614624, 4068224, 4096575, 4597424, 4975424, 6591375, 7574175, 8555624, 9511424, 10446975, 10749375, 10872224, 11477024, 12535424, 13773375, 13946624, 14277375, 15926624, 16041375, 16505775, 16769024

Offset: 1

Amiram Eldar, Sep 30 2019

nonn

			165375 is in the sequence since both 165375 and 165376 are nonunitary abundant: nusigma(165375) = 179280 > 165375, and nusigma(165376) = 183600 > 165376 (nusigma is the sum of nonunitary divisors, A048146).

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

Cf. A048146, A064597, A094889, A307823.

Cf. A096399, A292704, A318167, A327635.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); nuabQ[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (1 + Power @@@ FactorInteger[n]) > n; s = {}; q1 = False; Do[q2 = nuabQ[n]; If[q1 && q2, AppendTo[s, n - 1]]; q1 = q2, {n, 2, 10^7}]; s

A348275 Odd noninfinitary abundant numbers: the odd terms of A348274.

Original entry on oeis.org

99225, 1091475, 1289925, 1334025, 1576575, 1686825, 1715175, 1863225, 1885275, 2027025, 2061675, 2282175, 2304225, 2395575, 2401245, 2436525, 2480625, 2650725, 2723175, 2789325, 2877525, 2962575, 3031875, 3075975, 3132675, 3185325, 3186225, 3296475, 3353805, 3501225

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

The number of terms below 10^k, for k = 5, 6, ..., are 1, 113, 630, 7771, 73685, ... Apparently this sequence has an asymptotic density 0.000007...

			99225 is a term since A348271(99225) = 107207 > 99225.

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

Cf. A348271.
Subsequence of A005231 and A348274.
Similar sequences: A094889, A127666, A129485, A293186, A321147.

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]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[1, 2*10^6, 2], s[#] > # &]

A333950 Odd recursive abundant numbers: odd numbers k such that A333926(k) > 2*k.

Original entry on oeis.org

1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 8085, 8415, 8925, 9135, 9555, 9765, 11025, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 17325, 18585, 19215, 19635, 20475, 21105, 21945, 22365, 22995, 23205, 24255, 24885, 25935, 26145, 26565, 26775

Offset: 1

Amiram Eldar, Apr 11 2020

nonn

			1575 is a term since it is odd and A333926(1575) = 3224 > 2 * 1575.

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

Intersection of A005408 and A333928.
Cf. A333926.
Analogous sequences: A005231, A094889 (nonunitary), A129485 (unitary), A127666 (infinitary), A293186 (bi-unitary), A321147 (exponential).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); Select[2*Range[15000] + 1, recDivSum[#] > 2*# &]

A360526 Odd numbers k such that A360522(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, Feb 10 2023

nonn

First differs from A112643, A129485, A249263 at n = 46: a(46) = 165165 is not a term of these sequences.

			15015 is a term since A360522(15015) = 32256 > 2*15015.

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

Cf. A360522.
Subsequence of A005101, A005231 and A360525.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A348275, A348605.

f[p_, e_] := p^e + e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := s[n] > 2*n; Select[Range[1, 10^5, 2], q]

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

A339936 Odd coreful abundant numbers: the odd terms of A308053.

Original entry on oeis.org

99225, 165375, 231525, 297675, 496125, 694575, 826875, 893025, 1091475, 1157625, 1225125, 1289925, 1488375, 1620675, 1686825, 1819125, 1885275, 2083725, 2149875, 2282175, 2480625, 2546775, 2679075, 2811375, 2877525, 3009825, 3075975, 3142125, 3274425, 3472875

Offset: 1

Amiram Eldar, Dec 23 2020

nonn

The asymptotic density of this sequence is Sum_{n>=1} f(A356871(n)) = 9.1348...*10^(-6), where f(n) = (4/(Pi^2*n))*Product_{prime p|n}(p/(p+1)) if n is odd and 0 otherwise. - Amiram Eldar, Sep 02 2022

			99225 is a term since it is odd and the sum of its coreful divisors is A057723(99225) = 201600 > 2 * 99225.

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

Intersection of A005408 and A308053.
Subsequence of A321147.
Cf. A057723, A356871.
Similar sequences: A005231, A094889, A112643, A127666, A129485, A293186, A321147, A333950.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[1, 10^6, 2], s[#] > 2*# &]

A339938 Odd non-coreful abundant numbers: the odd terms of A308127.

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, 75075, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 105105

Offset: 1

Amiram Eldar, Dec 23 2020

nonn

First differs from A112643, A129485 and A249263 at n = 28.

			15015 is a term since it is odd and the sum of its non-coreful divisors is A308135(15015) = 17241 > 15015.

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

Intersection of A005408 and A308127.
Cf. A308135.
Similar sequences: A005231, A094889, A112643, A127666, A129485, A249263, A293186, A321147, A333950.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; s[1] = 0; s[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); Select[Range[1, 10^5, 2], s[#] > # &]

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]

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);

A380932 Odd numbers k such that A380845(k) > 2*k.

Original entry on oeis.org

322245, 590205, 874665, 966735, 1934415, 2900205, 3224025, 3378375, 3869775, 4729725, 6081075, 6449625, 6818175, 7740495, 8783775, 8906625, 9029475, 9889425, 10135125, 10961685, 11609325, 11821425, 12900825, 13378365, 14189175, 15049125, 15481935, 15909075, 16253055

Offset: 1

Amiram Eldar, Feb 08 2025

The odd terms in A380929.

Analogous to odd abundant numbers (A005231) with A380845 instead of A000203.

			322245 is a term since it is odd, and A380845(322245) = 679582 > 2 * 322245 = 644490.

Amiram Eldar, Table of n, a(n) for n = 1..2306

Cf. A000203, A380845.
Intersection of A005408 and A380929.
Subsequence of A005231.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A339938, A348275, A360526, A379031.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1,10^6,2], q]

isok(k) = if(!(k % 2), 0, my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k);

A357607 Odd numbers k such that A162296(k) > 2*k.

Original entry on oeis.org

4725, 6615, 7875, 8505, 11025, 14175, 15435, 17325, 19845, 20475, 22275, 23625, 24255, 25515, 26775, 28665, 29925, 31185, 33075, 36225, 36855, 37125, 37485, 38115, 39375, 40425, 41895, 42525, 46305, 47775, 48195, 50715, 51975, 53235, 53865, 55125, 57915, 59535

Offset: 1

Amiram Eldar, Oct 06 2022

nonn

The least term that is not divisible by 3 is a(89047132) = 134785275625.

The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 4, 60, 640, 6650, 66044, 660230, 6604594, 66073470, ... . Apparently, the asymptotic density of this sequence exists and equals 0.000660... .

			4725 is a term since it is odd, and A162296(4725) = 9728 > 2*4725.

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

Cf. A162296.
Subsequence of A005231, A013929 and A357605.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A348275, A348605.

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[3, 60000, 2], q]

cp's OEIS Frontend

A327942 Numbers k such that both k and k+1 are nonunitary abundant numbers (A064597).

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A348275 Odd noninfinitary abundant numbers: the odd terms of A348274.

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A333950 Odd recursive abundant numbers: odd numbers k such that A333926(k) > 2*k.

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A360526 Odd numbers k such that A360522(k) > 2*k.

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A339936 Odd coreful abundant numbers: the odd terms of A308053.

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A339938 Odd non-coreful abundant numbers: the odd terms of A308127.

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

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

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