A068403 -id:A068403 - cp's OEIS frontend

A307112 Primitive 3-abundant numbers: Numbers k such that sigma(k) > 3k (A068403) all of whose proper divisors d are 3-deficient numbers having sigma(d) < 3d.

180, 420, 504, 660, 780, 1584, 1848, 1872, 1890, 2184, 2352, 2376, 2772, 2856, 3150, 3192, 3276, 4284, 4410, 4788, 4896, 5100, 5292, 5700, 5796, 6864, 6900, 6930, 7344, 7728, 8190, 8208, 8424, 9744, 10296, 10416, 10710, 10944, 11550, 11970, 12012, 12432, 12870

Offset: 1

Amiram Eldar, Mar 25 2019

nonn

Analogous to A071395 with abundancy index 3 instead of 2.

Paul Erdős and János Surányi, Topics in the Theory of Numbers, New York: Springer, 2003, p. 243.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, Primitive alpha-abundant numbers, Mathematics of Computation, Vol. 43, No. 167 (1984), pp. 263-270.
Paul Erdős, On additive arithmetical functions and applications of probability to number theory, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, Vol. 3 (1956), pp. 13-19.
Paul Erdős, Remarks on number theory. I: On primitive alpha-abundant numbers, Acta Arithmetica., Vol. 5, No. 1 (1959), pp. 25-33, alternative link.

Cf. A000203, A068403, A071395.

Select[Range@50000, DivisorSigma[1, #] > 3 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 3 # &, Most@ Divisors@ #] == 1 &] (* after Michael De Vlieger at A071395 *)

A329189 3-admirable numbers: 3-abundant numbers (A068403) k such that exists a proper divisor d of k such that sigma(k) - 2d = 3k, where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

180, 240, 360, 420, 504, 540, 600, 780, 1080, 1344, 1872, 1890, 2016, 2184, 2352, 2376, 2688, 3192, 3276, 3744, 4284, 4320, 4680, 5292, 5376, 5796, 6048, 6552, 7128, 7344, 7440, 8190, 9504, 10296, 10416, 13776, 14850, 18600, 19824, 19872, 20496, 21528, 22932

Offset: 1

Amiram Eldar, Nov 07 2019

nonn

Analogous to admirable numbers (A111592) as 3-perfect numbers (A005820) are analogous to perfect numbers (A000396).

The proper divisors of each term k can be added to a sum of 2*k with one divisor taken with a minus sign.

			180 is a term since its proper divisors can be added to 1 + 2 - 3 + 4 + 5 + 6 + 9 + 10 + 12 + 15 + 18 + 20 + 30 + 36 + 45 + 60 + 90 = 360 = 2 * 180, with one divisor, 3, taken with a minus sign.

Amiram Eldar, Table of n, a(n) for n = 1..810 (terms below 10^11)

Cf. A000203, A000396, A068403, A111592.

aQ[n_] := (ab = DivisorSigma[1, n] - 3 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2]; Select[Range[23000], aQ]

A005820 3-perfect (triply perfect, tri-perfect, triperfect or sous-double) numbers: numbers such that the sum of the divisors of n is 3n.

Original entry on oeis.org

120, 672, 523776, 459818240, 1476304896, 51001180160

Offset: 1

N. J. A. Sloane

These six terms are believed to comprise all 3-perfect numbers. - cf. the MathWorld link. - Daniel Forgues, May 11 2010
If there exists an odd perfect number m (a famous open problem) then 2m would be 3-perfect, since sigma(2m) = sigma(2)*sigma(m) = 3*2m. - Jens Kruse Andersen, Jul 30 2014
According to the previous comment from Jens Kruse Andersen, proving that this sequence is complete would imply that there are no odd perfect numbers. - Farideh Firoozbakht, Sep 09 2014
If 2 were prepended to this sequence, then it would be the sequence of integers k such that numerator(sigma(k)/k) = A017665(k) = 3. - Michel Marcus, Nov 22 2015
From  Antti Karttunen, Mar 20 2021, Sep 18 2021, (Start):
Obviously, any odd triperfect numbers k, if they exist, have to be squares for the condition sigma(k) = 3*k to hold, as sigma(k) is odd only for k square or twice a square. The square root would then need to be a term of A097023, because in that case sigma(2*k) = 9*k. (See illustration in A347391).
Conversely to Jens Kruse Andersen's comment above, any 3-perfect number of the form 4k+2 would be twice an odd perfect number. See comment in A347870.
(End)

			120 = 2^3*3*5;  sigma(120) = (2^4-1)/1*(3^2-1)/2*(5^2-1)/4 = (15)*(4)*(6) = (3*5)*(2^2)*(2*3) = 2^3*3^2*5 = (3) * (2^3*3*5) = 3 * 120. - _Daniel Forgues_, May 09 2010

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 120, p. 42, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chap.15, pp 82-5, Belin/Pour la Science, Paris 2000.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 142.
David Wells, "The Penguin Book of Curious and Interesting Numbers," Penguin Books, London, 1986, pages 135, 159 and 185.

Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Kevin A. Broughan and Qizhi Zhou, Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4, JIS 13 (2010) 10.1.5
Alfred Brousseau, Number Theory Tables, Fibonacci Association, San Jose, CA, 1973, p. 138.
Seth Colbert-Pollack, Judy Holdener, Emily Rachfal, and Yanqi Xu, A DIY Project: Construct Your Own Multiply Perfect Number!, Math Horizons, Vol. 28, pp. 20-23, February 2021.
Farideh Firoozbakht and Maximilian F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Achim Flammenkamp, The Multiply Perfect Numbers Page [This page contains a lot of useful information, but be careful, not all the statements are correct. For example, it appears to claim that the six terms of this sequence are known to be complete, which is not the case. - _N. J. A. Sloane_, Sep 10 2014]
James Grime and Brady Haran, The Six Triperfect Numbers, Numberphile video (2018).
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
Fred Helenius, Link to Glossary and Lists
Masao Kishore, Odd Triperfect Numbers, Mathematics of Computation, vol. 42, no. 165, 1984, pp. 231-233.
Gérard P. Michon, Multiperfect and hemiperfect numbers
Walter Nissen, Abundancy : Some Resources
N. J. A. Sloane & A. L. Brown, Correspondence, 1974
Eric Weisstein's World of Mathematics, Multiperfect Number
Eric Weisstein's World of Mathematics, Sous-Double
Wikipedia, Multiply perfect number, (section Triperfect numbers)

Cf. A000203, A000396, A007539, A017665, A019278, A027687, A046060, A046061, A068403, A075701, A097023, A171266, A259302, A259303, A306373, A326051, A326181, A329189, A335141, A335254, A347383, A347391.
Subsequence of the following sequences: A007691, A069085, A153501, A216780, A292365, A336458, A336461, A336745, and if there are no odd terms, then also of A334410.
Positions of 120's in A094759, 119's in A326200.

A005820:=n->`if`(numtheory[sigma](n) = 3*n, n, NULL): seq(A005820(n), n=1..6*10^5); # Wesley Ivan Hurt, Oct 15 2017

triPerfectQ[n_] := DivisorSigma[1, n] == 3n; A005820 = {}; Do[If[triPerfectQ[n], AppendTo[A005820, n]], {n, 10^6}]; A005820 (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)
Select[Range[10^6],DivisorSigma[1,#]==3#&] (* Harvey P. Dale, Jul 03 2023 *)

isok(n) = sigma(n, -1) == 3; \\ Michel Marcus, Nov 22 2015

a(n) = 2*A326051(n). [provided no odd triperfect numbers exist] - Antti Karttunen, Jun 13 2019

Wells gives the 6th term as 31001180160, but this is an error.

Edited by Farideh Firoozbakht and N. J. A. Sloane, Sep 09 2014 to remove some incorrect statements.

A347391 Numbers k such that sigma(k) is either their sibling in Doudna tree (A005940), or one of the sibling's descendants.

Original entry on oeis.org

3, 4, 5, 15, 20, 189, 945, 2125, 6375, 9261, 46305, 401625, 19679625

Offset: 1

Antti Karttunen, Aug 30 2021

Numbers k > 1 such that nearest common ancestor of k and sigma(k) in Doudna tree is the parent of k, and sigma(k) is not a descendant of k.
Any hypothetical odd term x in A005820 (triperfect numbers) would also be a member of this sequence. This is illustrated in the following diagram which shows how the neighborhood of such x would look like in the Doudna tree (A005940). If m (the parent of x, x = A003961(m), m = A064989(x)) is even, then x is a multiple of 3, while if m is odd, then 3 does not divide x. Because the abundancy index decreases when traversing leftwards in the Doudna tree, m must be a term of A068403. Both x and m would also need to be squares, by necessity.
.
               <--A003961--  m  ---(*2)--->
             .............../ \...............
            /                                 \
           /                                   \
          x                                     2m
         / \                                   / \
  etc.../   \.....2x         sigma(x) = 3x..../   \.....4m
                  / \                  / \             / \
               etc.  etc.           etc.  \           /  etc.
                                           \         /
                                           6x       9x = sigma(2x)
                                           / \     / \
                                         etc. \ etc.  etc.
                                               \
                                               12x = sigma(3x) if m odd.
.
From the diagram we also see that 2x would then need to be a term of A347392 (as well as that of A159907 and also in A074388, thus sqrt(x) should be a term of A097023), and furthermore, if x is not a multiple of 3 (i.e., when m is odd), then sigma(3*x) = 4*sigma(x) = 4*(3*x), thus 3*x = sigma(x) would be a term of A336702 (particularly, in A027687) and x would be a term of A323653.
Moreover, any odd square x in this sequence (for which sigma(x) would also be odd), would have an abundancy index of at least three (sigma(x)/x >= 3). See comments in A347383.
Note how 401625 = 6375 * 63 = 945 * 425, 46305 = 945 * 49, 9261 = 189 * 49, 6375 = 2125 * 3, 945 = 189 * 5 = 15 * 63 and 9261*2125 = 19679625. It seems that when the multiplicands are coprime, then they are both terms of this sequence, e.g. 2125 and 3, 189 and 5, 2125 and 9261.
From Antti Karttunen, Jul 10 2024: (Start)
Regarding the observation above, for two coprime odd numbers x, y, if both are included here because sigma(x) = 2^a * A064989(x) and sigma(y) = 2^b * A064989(y), then also their product x*y is included because in that case sigma(x*y) = 2^(a+b) * A064989(x*y).
Also, for two coprime odd numbers x, y, if both are included here because sigma(x) = A065119(i) * x and sigma(y) = A065119(j) * y, then also their product x*y is included because sigma(x*y) = A065119(k) * x*y, where A065119(k) = A065119(i)*A065119(j). The existence of such numbers (that would include odd triperfect and odd 6-perfect numbers, see A046061) is so far hypothetical, none is known.
It is not possible that the odd x is in this sequence if sigma(x) = k*A003961^e(x) and e = A061395(k)-2 >= 1.
Note that all odd terms < 2^33 here are some of the exponentially odd divisors of 19679625 (see A374199, also A374463 and A374464).
(End)
Question: from a(6) = 189 onward, are the rest of terms all in A347390?
Conjecture: sequence is finite.
If it exists, a(14) > 2^33.

			Sigma(3) = 4 is located as the sibling of 3 in the Doudna-tree (see the illustration in A005940), thus 3 is included in this sequence.
Sigma(4) = 7 is located as a grandchild of 3 (which is the sibling of 4) in the Doudna-tree, thus 4 is included in this sequence.
Sigma(5) = 6 is located as the sibling of 5 in the Doudna-tree, thus 5 is included in this sequence.
189 (= 3^3 * 7) is a term, as sigma(189) = 320, and 320 occurs as a descendant of 80 (which is the right sibling of 189) in the Doudna tree, as illustrated below:
.
             40
            /  \
   A003961 /    \ *2
          /      \
        189       80
        / \      / \
     etc   etc etc  160
                   / \
                 etc  320
                     / \
                   etc. etc.
.
945 (= 3^3 * 5 * 7) is a term, as sigma(945) = 1920, and 1920 occurs as a descendant of 240, which is the right sibling of 945 in the Doudna tree, as illustrated below:
            120
            /  \
   A003961 /    \ *2
          /      \
        945       240
        / \      / \
     etc   etc  etc  480
                   / \
                 etc  960
                     / \
                   etc. 1920
                        / \
                     etc. etc.

Positions of 1's in A347381.
Cf. A000203, A003961, A005940, A005820, A065119, A068403, A074388, A097023, A159907, A374199, A374463, A374464.
Cf. also A027687, A292583, A323653, A336702, A347383, A347390, A347392.

PARI
```
isA347391(n) = (1==A347381(n));
```

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
isA347391(n) = if(1==n,0,my(m=A252463(n), s=sigma(n)); while(s>m, if(s==n, return(0)); s = A252463(s)); (s==m));

A068404 Numbers k such that sigma(k) > 4*k.

Original entry on oeis.org

27720, 50400, 55440, 60480, 65520, 75600, 83160, 85680, 90720, 95760, 98280, 100800, 105840, 110880, 115920, 120120, 120960, 128520, 131040, 138600, 141120, 143640, 151200, 163800, 166320, 171360, 176400, 180180, 181440, 184800, 191520

Offset: 1

Benoit Cloitre, Mar 02 2002

nonn

This sequence is of positive density, see for example Davenport. The density is between 0.000176 and 0.004521 according to the McDaniel College link. - Charles R Greathouse IV, Sep 07 2012
From Amiram Eldar, Feb 13 2021: (Start)
Behrend (1933) found the bounds (0.00003, 0.025) for the asymptotic density.
Wall et al. (1972) found the bounds (0.0001, 0.0147).
Using Deléglise's method the upper bound for the density found by McDaniel College is 0.000679406. (End)

Harold Davenport, Über numeri abundantes, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6 (1933), pp. 830-837.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Felix Behrend, Über numeri abundantes II, Preuss. Akad. Wiss. Sitzungsber., Vol. 6 (1933), pp. 280-293; alternative link.
Marc Deléglise, Bounds for the Density of Abundant Integers, Experimental Mathematics, Vol. 7, No. 2 (1998), pp. 137-143.
Richard Laatsch, Measuring the Abundancy of Integers, Mathematics Magazine, Vol. 59, No. 2 (1986), pp. 84-92, alternative link.
Gordon L. Miller and Mary T. Whalen, Multiply Abundant Numbers, School Science and Mathematics, Volume 95, Issue 5 (May 1995), pp. 256-259.
Summer 2010 research group on Abundancy, Abundancy Bounds 2010, McDaniel College, 2010.
Charles R. Wall, Phillip L. Crews and Donald B. Johnson, Density Bounds for the Sum of Divisors Function, Mathematics of Computation, Vol. 26, No. 119 (1972), pp. 773-777; Errata, Vol. 31, No. 138 (1977), p. 616.

Cf. A068403, A001221, A215264.

Cf. A027687 (4-perfect numbers).

Select[Range[27720,9!,60], 4*#Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)

A001221(a(n)) >= 4 (Laatsch, 1986). - Amiram Eldar, Nov 07 2020

A215264 Numbers k such that sigma(k) > 5*k.

Original entry on oeis.org

122522400, 147026880, 183783600, 205405200, 220540320, 232792560, 245044800, 273873600, 294053760, 328648320, 367567200, 410810400, 428828400, 441080640, 465585120, 490089600, 492972480, 497296800, 514594080, 537213600, 547747200, 551350800, 563603040

Offset: 1

Donovan Johnson, Aug 07 2012

nonn

The asymptotic density of this sequence is > 1/a(1) ~ 8*10^(-9). Wall et al. (1972) proved that it is < 0.0122. - Amiram Eldar, Feb 13 2021

			sigma(122522400) = 614210688 and 614210688 > 5 * 122522400.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Richard Laatsch, Measuring the Abundancy of Integers, Mathematics Magazine, Vol. 59, No. 2 (1986), pp. 84-92, alternative link.
Gordon L. Miller and Mary T. Whalen, Multiply Abundant Numbers, School Science and Mathematics, Volume 95, Issue 5 (May 1995), pp. 256-259.
Charles R. Wall, Phillip L. Crews and Donald B. Johnson, Density Bounds for the Sum of Divisors Function, Mathematics of Computation, Vol. 26, No. 119 (1972), pp. 773-777; Errata, Vol. 31, No. 138 (1977), p. 616.

Cf. A000203, A001221, A023199, A068403, A068404.

for(n=122522400, 563603040, if(sigma(n)>5*n, print1(n ", ")))

A001221(a(n)) >= 6 (Laatsch, 1986). - Amiram Eldar, Nov 07 2020

A358413 Smallest 3-abundant number (sigma(x) > 3x) which is not divisible by any of the first n primes.

Original entry on oeis.org

180, 1018976683725, 5164037398437051798923642083026622326955987448536772329145127064375

Offset: 0

Jianing Song, Nov 14 2022

Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(0) = 2^2*3^2*5, a(1) = 3^3*5^2*7^2*11*13*17*19*23*29, and a(2) = 5^4*7^3*11^2*13^2*17*...*157 ~ 5.16404*10^66. a(3) = 7^3*11^3*13^2*17^2*19^2*23^2*29^2*31*...*569 ~ 2.54562*10^239 and a(4) = 11^3*13^3*17^2*...*47^2*53*...*1597 ~ 3.99515*10^688 are too large to display.

			a(1) = A119240(3) = 1018976683725 is the smallest 3-abundant odd number.
a(2) = A358412(3) = 5164037398437051798923642083026622326955987448536772329145127064375 is the smallest 3-abundant number that is coprime to 2 and 3.

Jianing Song, Table of n, a(n) for n = 0..4
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

Cf. A068403 (3-abundant numbers).
Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), this sequence (k=3), A358414 (k=4).
Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), A358412 (p=5), A358418 (p=7), A358419 (p=11).

A380847 Numbers k such that A380845(k) = 3*k.

Original entry on oeis.org

1800, 3720, 7560, 15240, 20832, 30600, 42336, 61320, 85344, 109320, 116040, 122760, 171360, 218760, 238920, 245640, 343392, 346440, 395880, 437640, 462600, 484680, 491400, 580680, 687456, 854760, 875400, 896520, 917880, 925320, 950520, 954120, 976200, 982920, 1011720

Offset: 1

Amiram Eldar, Feb 05 2025

Analogous to triperfect numbers (A005820) with A380845 instead of A000203.

All the terms are 3-abundant numbers (A068403), because A380845(k) <= A000203(k) with equality only when k is a power of 2, and powers of 2 are deficient numbers (A005100).

			1800 is a term since A380845(18) = 5400 = 3 * 1800.

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

Cf. A000203, A005100, A005820, A380845, A380846, A380848.

Subsequence of A068403.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] == 3*k]; Select[Range[10^6], q]

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

A340109 Coreful 3-abundant numbers: numbers k such that csigma(k) > 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

5400, 7200, 10800, 14400, 16200, 18000, 21168, 21600, 27000, 28800, 32400, 36000, 37800, 42336, 43200, 48600, 50400, 54000, 56448, 57600, 59400, 63504, 64800, 70200, 72000, 75600, 79200, 81000, 84672, 86400, 88200, 90000, 91800, 93600, 97200, 98784, 100800, 102600

Offset: 1

Amiram Eldar, Dec 28 2020

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).
Analogous to A068403 as A308053 is analogous to A005101.
Apparently, the least odd term in this sequence is 3^4 * 5^3 * 7^3 * 11^2 * 13^2 * 17^2 * 19^2 * 23^2 * 29^2 = 3296233276111741840875.
The asymptotic density of this sequence is Sum_{n>=1} f(A364991(n)) = 0.0004006..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)). - Amiram Eldar, Aug 15 2023

			5400 is a term since csigma(5400) = 16380 > 3 * 5400.

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

Subsequence of A308053.
Cf. A007947, A057723, A364991 (primitive terms).
Similar sequences: A068403, A285615, A293187, A300664, A307112, A328135.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], s[#] > 3*# &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);}
is(n) = s(n) > 3*n; \\ Amiram Eldar, Aug 15 2023

A357493 Numbers k such that s(k) = 3*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

480, 2688, 56304, 89400, 195216, 2095104, 9724032, 69441408, 1839272960, 5905219584

Offset: 1

Amiram Eldar, Oct 01 2022

Analogous to 3-perfect numbers (A005820) with nonsquarefree divisors.
Equivalently, numbers k such that A325314(k) = -2*k.
a(11) > 10^11, if it exists.
If k is one of the 6 known 3-perfect numbers, then 4*k is a term.

			480 is a term since A162296(480) = 1440 = 3*480.

Cf. A162296, A325314.
Subsequence of A013929 and A068403.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), A322609 (m=2), this sequence (m=3), A357494 (m=4).
Similar sequence: A005820.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) == 3*n]; Select[Range[2, 10^7], q]

cp's OEIS Frontend

A307112 Primitive 3-abundant numbers: Numbers k such that sigma(k) > 3k (A068403) all of whose proper divisors d are 3-deficient numbers having sigma(d) < 3d.

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A329189 3-admirable numbers: 3-abundant numbers (A068403) k such that exists a proper divisor d of k such that sigma(k) - 2*d = 3*k, where sigma(k) is the sum of divisors of k (A000203).

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A005820 3-perfect (triply perfect, tri-perfect, triperfect or sous-double) numbers: numbers such that the sum of the divisors of n is 3n.

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A347391 Numbers k such that sigma(k) is either their sibling in Doudna tree (A005940), or one of the sibling's descendants.

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A068404 Numbers k such that sigma(k) > 4*k.

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A215264 Numbers k such that sigma(k) > 5*k.

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PARI

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A358413 Smallest 3-abundant number (sigma(x) > 3x) which is not divisible by any of the first n primes.

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A380847 Numbers k such that A380845(k) = 3*k.

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A329189 3-admirable numbers: 3-abundant numbers (A068403) k such that exists a proper divisor d of k such that sigma(k) - 2d = 3k, where sigma(k) is the sum of divisors of k (A000203).