A321147 -id:A321147 - cp's OEIS frontend

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

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]

A336679 Odd exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller odd exponential abundant number.

Original entry on oeis.org

225450225, 385533225, 481583025, 538472025, 672624225, 985646025, 1150227225, 1566972225, 1685513025, 2105433225, 2679615225, 6485886225, 6554064825, 6933060225, 9150077475, 179678493225, 185601564225, 191620685025, 195686793225

Offset: 1

Amiram Eldar, Jul 30 2020

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

The corresponding values of esigma(k)/k are 2.148..., 2.112..., 2.099..., 2.085..., 2.072..., ...

The exponential version of A188263.
The odd version of A336254.
Subsequence of A321147.
Similar sequences: A335052, A335053, A335055.
Cf. A051377.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; seq = {}; rm = 3; Do[r = esigma[n]/n; If[r > 2 && r < rm, rm = r; AppendTo[seq, n]], {n, 1, 10^9, 2}]; seq

A336681 Odd exponential admirable numbers: the odd terms of A336680.

Original entry on oeis.org

6485886225, 71344748475, 110260065825, 123231838275, 125730522225, 149175383175, 162485579025, 185601564225, 188090700525, 191620685025, 195686793225, 201062472975, 239977790325, 265921335225, 278893107675, 304836652575, 343751969925, 395639059725, 434554377075

Offset: 1

Amiram Eldar, Jul 30 2020

nonn

Exponential admirable numbers that are odd are relatively rare: there are 5742336 even exponential admirable numbers that are smaller than the first odd term, i.e., a(1) = A336680(5742337).

			6485886225 is a term since 6485886225 = 80535 + 241605 + ... + (-8456175) + ... + 2161962075 is the sum of its proper exponential divisors with one of them, 8456175, taken with a minus sign.

Amiram Eldar, Table of n, a(n) for n = 1..65

The exponential version of A109729.
Intersection of A005408 and A336680.
Subsequence of A321147.
Similar sequences: A329188, A334973, A334975.

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[1, 10^9, 2], expAdmQ]

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

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A336679 Odd exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller odd exponential abundant number.

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A336681 Odd exponential admirable numbers: the odd terms of A336680.

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Mathematica