A068403 -id:A068403 - cp's OEIS frontend

A374779 Numbers whose divisors have a mean abundancy index that is larger than 3.

10886400, 13305600, 14515200, 18144000, 19958400, 21772800, 23587200, 23950080, 24192000, 25401600, 26611200, 27216000, 29030400, 29937600, 30481920, 31449600, 31933440, 32659200, 33264000, 33868800, 35380800, 35925120, 36288000, 37739520, 38102400, 39312000, 39916800

Offset: 1

Amiram Eldar, Jul 19 2024

nonn

Numbers k such that A374777(k)/A374778(k) > 3.
The numbers whose mean abundancy index of divisors is larger than 2 are in A245214.
The least odd term in this sequence is 84712751711029943302437712454902728115050897458369518458984375.

			10886400 is a term since A374777(10886400)/A374778(10886400) = 70644571/23514624 = 3.004... > 3.

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

Subsequence of A245214.

Cf. A068403, A374777, A374778.

f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[4*10^7], s[#] > 3 &]

is(n) = {my(f = factor(n)); prod(i = 1, #f~, p=f[i,1]; e=f[i,2]; (-2*p - e*p + p^2 + e*p^2 + p^(-e))/((e + 1)*(p - 1)^2)) > 3;}

A067766 Numbers k such that sigma(k)^2 > 4*sigma_2(k) where sigma_2(k) is the sum of squares over the divisors of k.

Original entry on oeis.org

24, 36, 48, 60, 72, 84, 90, 96, 108, 120, 126, 132, 140, 144, 150, 156, 160, 168, 180, 192, 204, 210, 216, 228, 240, 252, 264, 270, 276, 280, 288, 300, 312, 320, 324, 330, 336, 360, 378, 384, 390, 396, 400, 408, 420, 432, 440, 450, 456, 462, 468, 480, 504

Offset: 1

Benoit Cloitre, Apr 04 2002

From Amiram Eldar, Apr 19 2025: (Start)
All the terms are abundant numbers: if k is a term then sigma(k) > 2 * sqrt(sigma_2(k)) >= 2 * sqrt(k^2) = 2*k.
All the 3-abundant numbers (A068403) are terms because sigma_2(k) < zeta(2) * k^2, so 2 * sqrt(sigma_2(k))/k < 2*sqrt(zeta(2)) = 2.565... < 3.
The numbers of terms that do not exceed 10^k, for k = 2, 3, ..., are 8, 109, 1110, 10874, 107610, 1085715, 10872432, 108442685, 1084358031, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1084... . (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000203, A001157.
Subsequence of A005101.
A068403 is a subsequence.

Select[Range[600],DivisorSigma[1,#]^2>4*DivisorSigma[2,#]&] (* Harvey P. Dale, Dec 27 2015 *)

for(n=1,1000,if(sigma(n)^2>4*sumdiv(n,k,k^2),print1(n,",")))

isok(k) = {my(f = factor(k)); sigma(f)^2 > 4 * sigma(f, 2);} \\ Amiram Eldar, Apr 19 2025

A333054 Numbers m such that r(m) > r(k) for all k < m, where r(m) = min(sigma(m)/m, sigma(m+1)/(m+1)), and sigma(m) is the sum of divisors of m (A000203).

Original entry on oeis.org

1, 2, 8, 14, 44, 104, 495, 944, 4095, 5775, 5984, 21735, 98175, 862784, 4096575, 7194824, 14753024, 879207615, 1969789184, 2275962975, 3968862975, 12567844575, 39566665215, 44803620225, 77510285775, 125617830975, 162902829375

Offset: 1

Amiram Eldar, Mar 06 2020

The corresponding values of r(a(n)) are 1, 1.333..., 1.444..., 1.6, 1.733..., 1.828..., 1.890..., 1.970..., 1.999..., 2.044..., 2.085..., 2.120..., 2.181..., 2.243..., 2.248..., 2.252..., 2.360..., 2.397..., 2.407..., 2.408..., 2.411...
The least number m such that both m and m+1 are k-abundant (i.e., their abundancy indices sigma(m)/m > k and sigma(m+1)/(m+1) > k) is a term in this sequence. E.g., a(10) = 5775 = A096399(1).
a(28) > 5*10^11. - Amiram Eldar, Jan 02 2021

			The values of min(sigma(k)/k, sigma(k+1)/(k+1)) for k = 1, 2, ... 8 are 1, 4/3, 4/3, 6/5, 6/5, 8/7, 8/7, 13/9. The record values in this range, 1, 4/3 and 13/9, are obtained at k = 1, 2, and 8.

Cf. A000203, A004394, A068403, A096399, A333053.

seq={}; rminmax = 0; r1 = 1; Do[r2 = DivisorSigma[1, n]/n; rmin = Min[r1, r2]; If[rmin > rminmax, rminmax = rmin; AppendTo[seq, n-1]]; r1 = r2, {n, 2, 10^6}]; seq

a(22)-a(27) from Amiram Eldar, Jan 02 2021

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 > # &]

A387153 Squarefree 3-abundant numbers: squarefree numbers k such that A000203(k) > 3*k.

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 79170, 82110, 84630, 85470, 91770, 94710, 99330, 101010, 103530, 108570, 111930, 117390, 122430, 128310, 136290, 140910, 144690, 154770, 161070, 164010, 166530, 168630, 182490, 191730, 205590

Offset: 1

Amiram Eldar, Aug 19 2025

nonn

First differs from A067885 at n = 11: A067885(11) = 72930 is not a term of this sequence. a(59) = 510510 is the least term of this sequence that is not in A067885.
Subsequence of A285615 and first differs from it at n = 51: A285615(51) = 390390 is not a term of this sequence.
This sequence is not the same as the sequence of numbers k such that A048250(k) > 3*k which includes all the terms of this sequence but also nonsquarefree numbers, the least of them is 2*A002110(52) = A088860(52) = 2.1248...*10^96.
The least odd term is A002110(17)/2 = 961380175077106319535, the least term that is not divisible by 3 is a(5607800) = 66853496710, and the least term that is coprime to 6 is A002110(52)/6 = 1.7706...*10^95.
If k is a term and m is a squarefree number coprime to k, then k*m is also a term.
The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 17, 95, 795, 8162, 86331, 854164, 8372782, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00008... .

			30030 = 2 * 3 * 5 * 7 * 11 * 13 is a term since it is squarefree, and sigma(30030) = 96768 > 3*30030 = 90090.

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

Intersection of A005117 and A068403.
Subsequence of A087248 and A285615.
Cf. A000203, A001221, A002110, A013929, A048250, A088860.

q[k_] := Module[{f = FactorInteger[k]}, Max[f[[;;, 2]]] == 1 && Times @@ (1 + f[[;; , 1]]) > 3*k]; Select[Range[2*10^5], q]

isok(k) = {my(f = factor(k)); issquarefree(f) && vecprod(apply(x -> x+1, f[, 1])) > 3*k;}

A001221(a(n)) >= 6.

A330245 Numbers m with a unique subset of the divisors of m that sums to m (A064771) such that sigma(m)/m > sigma(k)/k for all smaller terms k < m of A064771, where sigma(m) is the sum of divisors of m (A000203).

Original entry on oeis.org

6, 20, 78, 1014, 3774, 9514254

Offset: 1

Amiram Eldar, Dec 06 2019

Paul Erdős asked whether there are extra-weird numbers n, i.e., numbers n for which sigma(n)/n > 3, but n is not the sum of a subset of its divisors in two ways. Such numbers, if they exist, are in the intersection of A064771 and A068403, and the least of them is a term of this sequence.
a(6) > 2*10^5.
10^11 < a(7) <= 105590246974194. - Giovanni Resta, Jan 14 2020

			The abundancy indices of the terms are sigma(a(n))/a(n) = 2 < 2.1 < 2.153... < 2.165... < 2.174... < 2.1757...

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, p. 77.

Cf. A000203, A064771, A068403.

okQ[n_] := Module[{d = Most[Divisors[n]]}, SeriesCoefficient[Series[ Product[ 1+x^i, {i, d}], {x, 0, n}], n] == 1]; seq = {}; rm = 0; Do[If[(r = DivisorSigma[1, n]/n) > rm && okQ[n], rm = r; AppendTo[seq, n]], {n, 1, 4000}]; seq (* after Harvey P. Dale at A064771 *)

a(6) from Giovanni Resta, Jan 14 2020

cp's OEIS Frontend

A374779 Numbers whose divisors have a mean abundancy index that is larger than 3.

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A067766 Numbers k such that sigma(k)^2 > 4*sigma_2(k) where sigma_2(k) is the sum of squares over the divisors of k.

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A333054 Numbers m such that r(m) > r(k) for all k < m, where r(m) = min(sigma(m)/m, sigma(m+1)/(m+1)), and sigma(m) is the sum of divisors of m (A000203).

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A348627 Numbers that are both exponential and nonexponential abundant numbers.

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A387153 Squarefree 3-abundant numbers: squarefree numbers k such that A000203(k) > 3*k.

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A330245 Numbers m with a unique subset of the divisors of m that sums to m (A064771) such that sigma(m)/m > sigma(k)/k for all smaller terms k < m of A064771, where sigma(m) is the sum of divisors of m (A000203).

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