A129485 -id:A129485 - cp's OEIS frontend

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

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]

A294026 Odd unitary abundant numbers with a record small gap to the next odd unitary abundant number.

Original entry on oeis.org

15015, 19635, 21945, 25935, 33495, 1752135, 1915095, 1915305, 119104635, 134877405

Offset: 1

Amiram Eldar, Oct 22 2017

The corresponding gaps are 4620, 2310, 1260, 630, 420, 330, 210, 180, 120, 30.
The upper ends are 19635, 21945, 23205, 26565, 33915, 1752465, 1915305, 1915485, 119104755, 134877435, ...
The unitary version of A294025.
No more terms below 10^9.
10^13 < a(11) <= 42229304608764255 (gap 18), while t = 220730839027951785 and t+6 are a pair with gap 6. - Giovanni Resta, May 07 2020

			Odd unitary abundant numbers are 15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, ...
Their differences are 4620, 2310, 1260, 2730, 630, 4830, 2100, 420, ...
The records of small differences are 4620, 2310, 1260, 630, 420, ...
And the corresponding terms are 15015, 19635, 21945, 25935, 33495, ...

Cf. A129485, A294025.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; ouaQ[n_] := OddQ[n] && usigma[n] > 2 n; s = Select[Range[100000], ouaQ]; a={}; dmin = 5000; Do[d=s[[j+1]]-s[[j]]; If[d

usig(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d));
isok(n) = (n%2) && (usig(n) > 2*n);
lista(nn) = {last = 0; gap = oo; forstep(n=1, nn, 2, if (isok(n), if (last, if (n - last < gap, print1(last, ", "); gap = n - last)); last = n;););} \\ Michel Marcus, Dec 15 2017

cp's OEIS Frontend

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

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

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A294026 Odd unitary abundant numbers with a record small gap to the next odd unitary abundant number.

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