A068403 -id:A068403 - cp's OEIS frontend

A364977 Numbers k such that k/(3*k - sigma(k)) is a positive integer.

6, 24, 28, 60, 84, 168, 252, 270, 336, 496, 630, 756, 792, 864, 924, 936, 1140, 1170, 1488, 1638, 2268, 2808, 2970, 3672, 4464, 5148, 5472, 6804, 7308, 7644, 8128, 8700, 8910, 9300, 9936, 11172, 13392, 16368, 18018, 20196, 20412, 22230, 24384, 25116, 27888, 31968

Offset: 1

Amiram Eldar, Aug 15 2023

nonn

Analogous to A271816 as 3-abundant numbers (A068403) are analogous to abundant numbers (A005101).
Numbers k such that the sum of the divisors of k with one of them added twice is equal to 3*k.
The perfect numbers (A000396) are all terms.
For all the terms k, 2 <= sigma(k)/k < 3, i.e., they are all nondeficient numbers (A023196), and none are 3-abundant (A068403).

			6 is a term since 3*6 - sigma(6) = 6 > 0 and 6 is divisible by 6.
24 is a term since 3*24 - sigma(24) = 12 > 0 and 24 is divisible by 12.

Amiram Eldar, Table of n, a(n) for n = 1..600

Cf. A000203 (sigma), A000396, A023196, A068403, A271816, A329189, A364976.

Select[Range[32000], (d = 3*# - DivisorSigma[1, #]) > 0 && Divisible[#, d] &]

is(n) = {my(d = 3*n - sigma(n)); d > 0 && n%d == 0;}

A291457 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 3.

Original entry on oeis.org

180, 240, 360, 420, 480, 540, 600, 660, 780, 840, 1080, 1320, 1560, 1890, 1920, 2016, 2040, 2184, 2280, 2352, 2376, 2688, 2760, 2856, 3000, 3192, 3360, 3480, 3720, 3744, 4284, 4320, 4440, 4680, 4704, 4896, 4920, 5160, 5292, 5640, 5796, 6048, 6360, 6552, 7080, 7128

Offset: 1

Paolo P. Lava, Aug 24 2017

Case k=2 are the admirable numbers (A111592).

Subset of A023197, A068403, A068545, A204828, A204830.

			One of the proper divisors of 1080 is 120 and sigma(1080) - 3*120 = 3600 - 360 = 3240 = 3*1080.
One of the proper divisors of 17850 is 6 and sigma(17850) - 3*6 = 53568 - 18 = 53550 = 3*17850.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1500 from Paolo P. Lava)

Cf. A000203, A023197, A068403, A068545, A111592, A204828, A204830, A291458, A291459.

with(numtheory): P:=proc(q,h) local a,b,c,k; c:=0; a:=sort([op(divisors(q))]); for k from 1 to nops(a)-1 do if sigma(q)-h*a[k]=h*q then c:=1; break; fi; od; if c=1 then q; fi; end: seq(P(i,3),i=1..7200);

k=3; Select[Range[7128], (t = DivisorSigma[1, #]/k - #; # > t > 0 && IntegerQ[t] && Mod[#, t] == 0) &] (* Giovanni Resta, Aug 25 2017 *)

A380930 Numbers k such that A380845(k) > 3*k.

Original entry on oeis.org

1080, 2160, 3600, 4320, 7200, 7440, 8640, 11340, 13608, 14400, 14880, 15120, 17280, 18600, 22680, 22860, 27216, 28800, 29760, 30240, 30480, 31752, 33264, 34020, 34560, 37200, 41664, 45360, 45720, 45900, 51408, 53340, 54432, 57600, 59520, 60480, 60960, 61200, 63504

Offset: 1

Amiram Eldar, Feb 08 2025

Analogous to 3-abundant numbers (A068403) with A380845 instead of A000203.

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

Cf. A000203, A380845, A380846, A380847.
Subsequence of A068403 and A380929.
Subsequences: A380848, A380931.
Similar sequences: A285615, A293187, A300664, A328135, A340109.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 3*k]; Select[Range[64000], q]

isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 3*k;}

1080 is a term since A380845(1080) = 3330 > 3 * 1080 = 3240.

A067808 Numbers k such that sigma(k)^2 > 3*sigma(k^2).

Original entry on oeis.org

720, 1080, 1440, 1680, 1800, 2016, 2160, 2520, 2880, 3024, 3240, 3360, 3600, 3780, 3960, 4032, 4200, 4320, 4680, 5040, 5280, 5400, 5544, 5760, 6048, 6120, 6300, 6480, 6720, 6840, 7056, 7200, 7560, 7920, 8064, 8400, 8640, 9000, 9072, 9240, 9360, 9504

Offset: 1

Benoit Cloitre, Feb 07 2002

nonn

For every m>1 sigma(m)^2 > sigma(m^2).
From Robert Israel, Jun 20 2018: (Start)
Numbers with prime factorization Product_j p_j^(e_j) such that  Product_j (p_j^(e_j+1)-1)^2/((p_j^(2*e_j+1)-1)*(p_j-1)) > 3.
If h is a term then so are all multiples of h.
The first term that is squarefree is 7420738134810 = A002110(12). (End)
From Amiram Eldar, Apr 27 2025: (Start)
All the terms are 3-abundant numbers (A068403).
The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 1, 44, 501, 5246, 51870, 518782, 5191909, 51889993, 518783441, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00518... . (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002110, A065764.

Subsequence of A068403.

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  mul((t[1]^(t[2]+1)-1)^2/(t[1]^(2*t[2]+1)-1)/(t[1]-1), t = F) > 3
end proc:
select(filter, [$1..10^4]); # Robert Israel, Jun 20 2018

filterQ[n_] := Module[{F = FactorInteger[n]}, If[n == 1, Return[False]]; Product[{p, e} = pe; (p^(e+1)-1)^2/((p^(2e+1)-1)(p-1)), {pe, F}] > 3];
Select[Range[10^4], filterQ] (* Jean-François Alcover, Apr 29 2019, after Robert Israel *)

isok(k) = sigma(k)^2 > 3*sigma(k^2); \\ Michel Marcus, Apr 29 2019

isok(k) = {my(f = factor(k), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(e+1)-1)^2/((p^(2*e+1)-1)*(p-1))) > 3;} \\ Amiram Eldar, Apr 27 2025

A126104 Numbers n not divisible by 6 such that sigma(n) > 3n.

Original entry on oeis.org

40040, 61600, 72800, 80080, 104720, 117040, 123200, 123760, 138320, 141680, 145600, 154000, 160160, 172480, 200200, 209440, 215600, 234080, 246400, 247520, 254800, 261800, 276640, 280280, 283360, 291200, 292600, 308000

Offset: 1

Ant King, Mar 07 2007

nonn

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

Intersection of A047253 and A068403.

Select[Range[310000], Mod[#, 6] > 1 && DivisorSigma[1, #] > 3# &] (* Amiram Eldar, Dec 03 2019 *)

A259312 n such that 3 < sigma(n)/n < sigma(m)/m for all abundant numbers m

Original entry on oeis.org

180, 780, 1872, 2352, 16830, 17850, 20496, 51060, 59724, 64890, 430272, 884730, 3767100, 4946508, 15287976, 33806052, 34747416, 40974318, 137168580, 140492772, 157048290, 184773312, 3615557148, 16709332608, 40564903620, 40936559976, 60782804964

Offset: 1

Michel Marcus, Jun 24 2015

nonn

This sequence is similar to A071927, but with ratio 3 rather than 2.

a(28) > 7*10^10. - Giovanni Resta, Jun 24 2015

Cf. A068403 (sigma(n)>3n), A071927 (barely abundant).

lista(nn) =  {abk = 4; for (n = 1, nn, ab = sigma(n)/n; if ((ab > 3) && (ab < abk), print1(n, ", "); abk = ab););}

a(23)-a(27) from Giovanni Resta, Jun 24 2015

A335030 Numbers m that are not practical and have an abundancy index sigma(m)/m which is larger than that of any smaller number that is not practical.

Original entry on oeis.org

3, 9, 10, 44, 70, 102, 350, 372, 1608, 3492, 6096, 10380, 44040, 100260, 180240, 425160, 1744560, 2425080, 5509980, 10048080, 23614920, 97639920, 396315360, 900229680, 2519017200, 3113704440, 12870562320, 52307529120

Offset: 1

Amiram Eldar, May 20 2020

None of the terms are superabundant (A004394) since all the superabundant numbers are practical numbers (A005153).
The least term m that is k-abundant (having sigma(m)/m > k) for k = 2, 3, ... is A005101(14) = 70, A068403(896) = 44040, A068404(792087) = 3113704440, ...
What is the least 5-abundant number (A215264) that is not practical?

			The first 5 numbers that are not practical are m = 3, 5, 7, 9, 10. Their abundancy indices sigma(m)/m are 1.333..., 1.2, 1.142..., 1.444..., 1.8. The record values occur at 3, 9 and 10.

Cf. A000203 (sigma), A004394, A005153, A237287, A330244, A335029.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[fct_] := (ind = Position[fct[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}; seq = {}; rm = 1; Do[fct = FactorInteger[n]; r = Times@@((First/@fct^ (1+Last/@ fct)-1)/(First/@fct-1))/n; If[r > rm && !pracQ[fct], rm = r; AppendTo[seq, n]], {n, 3, 10^5}]; seq

A335141 Numbers that are both unitary pseudoperfect (A293188) and nonunitary pseudoperfect (A327945).

Original entry on oeis.org

840, 2940, 7260, 9240, 10140, 10920, 13860, 14280, 15960, 16380, 17160, 18480, 19320, 20580, 21420, 21840, 22440, 23100, 23940, 24024, 24360, 25080, 26040, 26520, 27300, 28560, 29640, 30360, 30870, 31080, 31920, 32340, 34440, 34650, 35700, 35880, 36120, 36960

Offset: 1

Amiram Eldar, May 25 2020

nonn

All the terms are nonsquarefree (since squarefree numbers do not have nonunitary divisors).

All the terms are either 3-abundant numbers (A068403) or 3-perfect numbers (A005820). None of the 6 known 3-perfect numbers are terms of this sequence. If there is a term that is 3-perfect, it is also a unitary perfect (A002827) and a nonunitary perfect (A064591).

			840 is a term since its aliquot unitary divisors are {1, 3, 5, 7, 8, 15, 21, 24, 35, 40, 56, 105, 120, 168, 280} and 1 + 5 + 7 + 8 + 15 + 35 + 40 + 56 + 105 + 120 + 168 + 280 = 840, and its nonunitary divisors are {2, 4, 6, 10, 12, 14, 20, 28, 30, 42, 60, 70, 84, 140, 210, 420} and 70 + 140 + 210 + 420 = 840.

Amiram Eldar, Table of n, a(n) for n = 1..400

Intersection of A293188 and A327945.
Subsequence of A335140.
Cf. A002827, A005820, A064591, A068403.

pspQ[n_] := Module[{d = Divisors[n], ud, nd, x}, ud = Select[d, CoprimeQ[#, n/#] &]; nd = Complement[d, ud]; ud = Most[ud]; Plus @@ ud >= n && Plus @@ nd >= n && SeriesCoefficient[Series[Product[1 + x^ud[[i]], {i, Length[ud]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^nd[[i]], {i, Length[nd]}], {x, 0, n}], n] > 0]; Select[Range[10^4], pspQ]

A348523 Numbers that are both infinitary and noninfinitary abundant numbers.

Original entry on oeis.org

960, 1440, 1800, 2016, 2400, 2940, 3240, 3528, 3780, 4536, 4860, 6720, 7260, 8640, 10080, 10140, 10560, 12096, 12480, 12600, 13860, 14784, 15120, 15360, 15840, 16320, 16380, 16800, 17472, 17640, 18240, 18480, 18720, 18900, 19008, 19800, 20160, 21420, 21600, 21840

Offset: 1

Amiram Eldar, Oct 21 2021

nonn

Apparently, the smallest odd term is 9170790153525.

			960 is a term since A049417(960) = 2040 > 2*960 = 1920 and A348271(960) = 1008 > 960.

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

Intersection of A129656 and A348274.
Subsequence of A068403.
Cf. A049417, A348271.

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]; q[n_] := (i = isigma[n]) > 2*n && DivisorSigma[1, n] - i > n; Select[Range[10^4], q]

A364976 3-abundant numbers k such that k/(sigma(k)-3*k) is an integer.

Original entry on oeis.org

180, 240, 360, 420, 540, 600, 780, 1080, 1344, 1872, 1890, 2016, 2184, 2352, 2376, 2688, 3192, 3276, 3744, 4284, 4320, 4680, 5292, 5376, 5796, 6048, 6552, 7128, 7440, 8190, 10416, 13776, 14850, 18600, 19824, 19872, 20496, 21528, 22932, 25056, 26208, 26496, 26784

Offset: 1

Amiram Eldar, Aug 15 2023

nonn

Analogous to A153501 as 3-abundant numbers (A068403) are analogous to abundant numbers (A005101).

Numbers k such that the sum of the divisors of k except for one of them is equal to 3*k.

			180 is a term since sigma(180) - 3*180 = 6 > 0 and 180 is divisible by 6.

Amiram Eldar, Table of n, a(n) for n = 1..576 (terms below 2.5*10^10)

Subsequence of A068403.
A027687 is a subsequence.
Cf. A000203 (sigma), A005101, A005820, A055153, A153501, A160320, A218404, A218406, A329189, A364977.

Select[Range[27000], (d = DivisorSigma[1, #] - 3*#) > 0 && Divisible[#, d] &]

is(n) = {my(d = sigma(n) - 3*n); d > 0 && n%d == 0;}

cp's OEIS Frontend

A364977 Numbers k such that k/(3*k - sigma(k)) is a positive integer.

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A291457 Numbers n having a proper divisor d such that sigma(n) - k*d = k*n. Case k = 3.

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A380930 Numbers k such that A380845(k) > 3*k.

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A067808 Numbers k such that sigma(k)^2 > 3*sigma(k^2).

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A126104 Numbers n not divisible by 6 such that sigma(n) > 3n.

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A259312 n such that 3 < sigma(n)/n < sigma(m)/m for all abundant numbers m

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A335030 Numbers m that are not practical and have an abundancy index sigma(m)/m which is larger than that of any smaller number that is not practical.

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A335141 Numbers that are both unitary pseudoperfect (A293188) and nonunitary pseudoperfect (A327945).

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A291457 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 3.