A023197 -id:A023197 - cp's OEIS frontend

A033880 Abundance of n, or (sum of divisors of n) - 2n.

-1, -1, -2, -1, -4, 0, -6, -1, -5, -2, -10, 4, -12, -4, -6, -1, -16, 3, -18, 2, -10, -8, -22, 12, -19, -10, -14, 0, -28, 12, -30, -1, -18, -14, -22, 19, -36, -16, -22, 10, -40, 12, -42, -4, -12, -20, -46, 28, -41, -7, -30, -6, -52, 12, -38, 8, -34, -26, -58, 48, -60, -28, -22

Offset: 1

N. J. A. Sloane

For no known n is a(n) = 1. If there is such an n it must be greater than 10^35 and have seven or more distinct prime factors (Hagis and Cohen 1982). - Jonathan Vos Post, May 01 2011
a(n) = -1 iff n is a power of 2. a(n) = 1 - n iff n is prime. - Omar E. Pol, Jan 30 2014 [If a(n) = -1 then n is called a least deficient number or an almost perfect number. All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number that is not a power of 2. See A000079. - Jianing Song, Oct 13 2019]
According to Deléglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480 (cf. A302991). Since the perfect numbers having density 0, the deficient numbers have density 0.7520 < 1 - A(2) < 0.7526 (cf. A318172). - Daniel Forgues, Oct 10 2015
2-abundance of n, a special case of the k-abundance of n, defined as (sum of divisors of n) - k*n, k >= 1. - Daniel Forgues, Oct 24 2015
Not to be confused with the abundancy of n, defined as (sum of divisors of n) / n. (Cf. A017665 / A017666.) - Daniel Forgues, Oct 25 2015

			For n = 10 the divisors of 10 are 1, 2, 5, 10. The sum of proper divisors of 10 minus 10 is 1 + 2 + 5 - 10 = -2, so the abundance of 10 is a(10) = -2. - _Omar E. Pol_, Dec 27 2013

Richard K. Guy, "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." Section B2 in Unsolved Problems in Number Theory, 2nd ed., New York: Springer-Verlag, pp. 45-53, 1994.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

J. G. Wurtzel, Table of n, a(n) for n = 1..10000 [This replaces an earlier b-file computed by T. D. Noe]
Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.
Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), 137-143.
Peter Hagis Jr. and Graeme L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
Eric Weisstein's World of Mathematics, Abundance.
Eric Weisstein's World of Mathematics, Abundancy.
Eric Weisstein's World of Mathematics, Quasiperfect Number.

Equals -A033879.
Cf. A000203, A001065. A005843.
Lists of positions where certain values occur: A005100 (a(n) < 0), A000396 (a(n) = 0) and A005101 (a(n) > 0), A023197 (a(n) >= n), A028982 (a(n) odd).
Cf. A072691, A302991, A318172.

[SumOfDivisors(n)-2*n: n in [1..100]]; // Vincenzo Librandi, Oct 11 2015

Maple
```
with(numtheory); n->sigma(n) - 2*n;
```

Array[Total[Divisors[#]]-2#&,70] (* Harvey P. Dale, Sep 16 2011 *)
Table[DivisorSigma[1, n] - 2*n, {n, 1, 70}] (* Amiram Eldar, Jun 09 2022 *)

a(n)=sigma(n)-2*n \\ Charles R Greathouse IV, Nov 20 2012

from sympy import divisor_sigma
def A033880(n): return divisor_sigma(n)-(n<<1) # Chai Wah Wu, Apr 12 2024

[sigma(n, 1)-2*n for n in range(1, 64)] # Stefano Spezia, Jul 18 2025

a(n) = A000203(n) - A005843(n). - Omar E. Pol, Dec 14 2008
a(n) = A001065(n) - n. - Omar E. Pol, Dec 27 2013
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/12 - 1 = -0.1775329665... . - Amiram Eldar, Apr 06 2024

Definition corrected Jul 04 2005

A068403 Numbers k such that sigma(k) > 3*k.

Original entry on oeis.org

180, 240, 360, 420, 480, 504, 540, 600, 660, 720, 780, 840, 900, 960, 1008, 1080, 1200, 1260, 1320, 1344, 1440, 1512, 1560, 1584, 1620, 1680, 1800, 1848, 1872, 1890, 1920, 1980, 2016, 2040, 2100, 2160, 2184, 2280, 2340, 2352, 2376, 2400, 2520, 2640

Offset: 1

Benoit Cloitre, Mar 02 2002

Davenport shows that these numbers have positive density. Are there good bounds for the density?
G. Miller & M. Whalen suggested that 1018976683725 (3^3*5^2*7^2*11*13*17*19*23*29) might be the smallest odd number in the sequence (a fact now, see A119240 and A023197). - Michel Marcus, May 01 2013
From Amiram Eldar, Feb 13 2021: (Start)
Behrend (1933) found the bounds (0.009, 0.110) for the asymptotic density.
Wall et al. (1972) found the bounds (0.0186, 0.0461).
The upper bound was reduced to 0.0214614 using Deléglise's method by McDaniel College (2010). (End)
Note that 1018976683725, the smallest odd term in this sequence, is A053624(51). - Charles R Greathouse IV, Jan 09 2025

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.

Terms not divisible by 6 are in A126104.
Cf. A001221, A068404, A215264.
Cf. A005820 (3-perfect numbers).

A068403:=n->`if`((numtheory)[sigma](n) > 3*n, n, NULL): seq(A068403(n), n=1..5*10^3); # Wesley Ivan Hurt, Apr 09 2017

Select[Range[180, 2000], 3*# < Plus@@Divisors[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)
Select[Range[3000],DivisorSigma[1,#]>3#&] (* Harvey P. Dale, Aug 12 2023 *)

for(n=1, 3000, if(sigma(n)>3*n, print1(n,", "))) \\ Indranil Ghosh, Apr 10 2017

from sympy import divisor_sigma
print([n for n in range(180, 3001) if divisor_sigma(n)>3*n]) # Indranil Ghosh, Apr 10 2017

A001221(a(n)) >= 3 (Laatsch, 1986). - Amiram Eldar, Nov 07 2020

a(n) ~ k*n for some constant k with 46 < k < 54. - Charles R Greathouse IV, Jan 21 2025

A204830 Numbers k whose divisors can be partitioned into three disjoint sets whose sums are all sigma(k)/3.

Original entry on oeis.org

120, 180, 240, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 960, 1080, 1260, 1320, 1440, 1512, 1560, 1584, 1620, 1680, 1800, 1848, 1890, 1920, 1980, 2016, 2040, 2160, 2184, 2280, 2340, 2352, 2376, 2400, 2520, 2640, 2688, 2760, 2772, 2856, 2880, 2940, 3000

Offset: 1

Jaroslav Krizek, Jan 22 2012

nonn

Subsequence of the intersection of A023197 and A087943.
If m is a term then so is m*p^k when p is coprime to m. - David A. Corneth, Mar 09 2024
Is this sequence equal to the sequence: "Numbers k such that sigma(k) is divisible by 3 and sigma(k) >= 3*k"? - David A. Corneth, Mar 17 2024
Answer: No. The numbers k with sigma(k) >= 3k and sigma(k) divisible by 3 that are not in this sequence are in A306476. - Amiram Eldar, Jun 22 2024

			180 is a term because sigma(180)/3 = 182 = 2 + 180 = 1+3+4+5+6+9+10+15+18+30+36+45 = 12+20+60+90 (summands are all the divisors of 180).

David A. Corneth, Table of n, a(n) for n = 1..10296 (first 500 terms from Paolo P. Lava, terms <= 550000).
Farid Jokar, On k-layered numbers, arXiv:2207.09053 [math.NT], 2022.

Cf. A023197, A083207 (Zumkeller numbers -- numbers k whose divisors can be partitioned into two disjoint sets whose sums are both sigma(k)/2), A087943, A204831 (numbers k whose divisors can be partitioned into four disjoint sets whose sums are all sigma(k)/4), A306476.

A103288 Numbers k such that sigma(k) >= 2k-1 (union of perfect, abundant and least deficient numbers).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 128, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220

Offset: 1

Max Alekseyev, Jan 28 2005

nonn

If the only least deficient numbers are the powers of 2 (open problem) then this sequence is the union of A023196 and A000079.

Like the abundant numbers, this sequence has density between 0.2474 and 0.2480, see A005101. - Charles R Greathouse IV, Nov 30 2022

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A023196, A023197, A023198, A023199, A000079, A004125.

Select[Range[500], DivisorSigma[1, #] >= 2*# - 1 &] (* Paolo Xausa, Dec 09 2024 *)

for(n=1,1000,if(sigma(n)>=2*n-1,print(n)));

Numbers k such that A004125(k) <= A004125(k-1).

A285615 Numbers k such that usigma(k) >= 3*k, where usigma(k) = sum of unitary divisors of k (A034448).

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

Offset: 1

Amiram Eldar, Apr 22 2017

nonn

Unitary 3-abundant numbers, correspond to 3-abundant numbers (A023197).
Similarly, the first numbers k such that usigma(k) >= 4*k are 200560490130, 7420738134810, 8222980095330, and 8624101075590. - Giovanni Resta, Apr 23 2017
The least odd term in this sequence is A070826(17) = 961380175077106319535 and the least odd number k such that usigma(k) >= 4*k is A070826(52) = 5.312...*10^95. - Amiram Eldar, Dec 26 2020

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A034448, A023197, A034683, A070826.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; Select[Range[100000], usigma[#] >= 3*# &]

isok(k) = sumdivmult(k, d, if(gcd(d, k/d)==1, d)) >= 3*k; \\ Michel Marcus, Dec 26 2020

A328135 Exponential 3-abundant numbers: numbers m such that esigma(m) >= 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

901800900, 1542132900, 1926332100, 2153888100, 2690496900, 2822796900, 3942584100, 4487660100, 4600908900, 5127992100, 6267888900, 6742052100, 7162236900, 7305120900, 8421732900, 8969984100, 9866448900, 10203020100, 10718460900, 11723411700, 11787444900, 12528324900

Offset: 1

Amiram Eldar, Oct 04 2019

nonn

Aiello et al. found bounds on e-multiperfect numbers, i.e., numbers m such that esigma(m) = k * m for k > 2: 2 * 10^7 for k = 3, and 10^85, 10^320, and 10^1210 for k = 4, 5, and 6. The data of this sequence raise the bound for exponential 3-perfect numbers to 3 * 10^10.
The least odd term is (59#/2)^2 = 924251841031287598942273821762233522616225. The least term which is coprime to 6 is (239#/6)^2 = 3.135... * 10^190.
The least exponential 4-abundant number (esigma(m) >= 4m) is (31#)^2 = 40224510201185827416900. In general, the least exponential k-abundant number (esigma(m) >= k*m), for k > 2, is (A002110(A072986(k)))^2.
The asymptotic density of this sequence is Sum_{n>=1} f(A383699(n)) = 1.325...*10^(-9), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, May 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
W. Aiello, G. E. Hardy, and M. V. Subbarao, On the existence of e-multiperfect numbers, Fibonacci Quarterly, Vol. 25, No. 1 (1987), pp. 65-71.

Subsequence of A129575.
A383699 is a subsequence.
Cf. A002110, A051377, A072986, A321147.
Cf. A023197, A307112, A285615 (unitary), A293187 (bi-unitary), A300664 (infinitary).

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^10], esigma[#] >= 3 # &]

A204828 Numbers n with abundancy 3 <= sigma(n)/n < 4.

Original entry on oeis.org

120, 180, 240, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1080, 1200, 1260, 1320, 1344, 1440, 1512, 1560, 1584, 1620, 1680, 1800, 1848, 1872, 1890, 1920, 1980, 2016, 2040, 2100, 2160, 2184, 2280, 2340, 2352, 2376, 2400, 2520, 2640

Offset: 1

Jaroslav Krizek, Jan 22 2012

nonn

A subsequence of A023197 (numbers n such that sigma(n) >= 3n) which is in turn a subsequence of the abundant numbers A005101, i.e., numbers n with sigma(n)/n > 2.
Differs from A023197 from a(565) on: The first term of A023197 which is not in this sequence is A023197(565) = 27720 = A023198(1) = A023199(4), the least number with abundancy >= 4.
Numbers with abundancy sigma(n)/n < 2 are called deficient and listed in A005100. Numbers with sigma(n)/n in the interval [2,3) are listed in A204829. Numbers with sigma(n)/n in the interval [4,5) are listed in A230608. - M. F. Hasler, Dec 05 2013

			Number 180 is in the sequence because sigma(180)/180 = 546/180 = 3.0333...

Jaroslav Krizek, Table of n, a(n) for n = 1..3000
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A204829 (abundant numbers with abundancy 2 <= a < 3).

A293187 Bi-unitary 3-abundant numbers: numbers k such that bsigma(k) > 3*k, where bsigma is the sum of the bi-unitary divisors function (A188999).

Original entry on oeis.org

480, 840, 1080, 1320, 1512, 1560, 1680, 1848, 1890, 1920, 2040, 2184, 2280, 2376, 2688, 2760, 2856, 3000, 3192, 3240, 3360, 3480, 3720, 3840, 4320, 4440, 4920, 5160, 5280, 5640, 5880, 6048, 6240, 6360, 6720, 7080, 7320, 7392, 7560, 7680, 8040, 8160, 8520

Offset: 1

Amiram Eldar, Oct 01 2017

nonn

Analogous to 3-abundant numbers (A023197) with bi-unitary sigma (A188999) instead of sigma (A000203).

			480 is in the sequence since bi-unitary sigma(480) = 1512 > 3 * 480.

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

Cf. A023197, A285615, A188999, A292982.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] :=  DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bAbundantQ[n_] := bsigma[n] > 3 n; Select[Range[1000], bAbundantQ] (* after Michael De Vlieger at A188999 *)

A300664 Infinitary 3-abundant numbers: numbers n such that isigma(n) >= 3n, where isigma is the sum of infinitary divisors of n (A049417).

Original entry on oeis.org

120, 840, 1080, 1320, 1512, 1560, 1848, 1890, 1920, 2040, 2184, 2280, 2376, 2688, 2760, 2856, 3000, 3192, 3480, 3720, 4440, 4920, 5160, 5640, 5880, 6360, 7080, 7320, 7560, 8040, 8520, 8760, 9240, 9480, 9720, 9960, 10680, 10920, 11640, 11880, 12120, 12360

Offset: 1

Amiram Eldar, Mar 10 2018

nonn

Analogous to 3-abundant numbers (A023197) with isigma (A049417) instead of sigma (A000203).

			840 is in the sequence since isigma(840) = 2880 > 3 * 840.

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

Cf. A007357, A023197, A049417, A129656, A129657.

ExponentList[n_Integer, factors_List] := {#, IntegerExponent[n, #]} & /@ factors; InfinitaryDivisors[1] := {1}; InfinitaryDivisors[n_Integer ? Positive] := Module[{factors = First /@ FactorInteger[n], d = Divisors[n]}, d[[Flatten[ Position[ Transpose[ Thread[Function[{f, g}, BitOr[f, g] == g][#, Last[#]]] & /@ Transpose[ Last /@ ExponentList[#, factors] & /@ d]], ?(And @@ # &), {1}]]]]]; properinfinitarydivisorsum[k] := Plus @@ InfinitaryDivisors[k] - k; Infinitary3AbundantNumberQ[k_] :=  If[properinfinitarydivisorsum[k] >= 2 k, True, False]; Select[Range[15000], Infinitary3AbundantNumberQ[#] &] (* after Ant King at A129656 *)

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 *)

cp's OEIS Frontend

A033880 Abundance of n, or (sum of divisors of n) - 2n.

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A068403 Numbers k such that sigma(k) > 3*k.

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A204830 Numbers k whose divisors can be partitioned into three disjoint sets whose sums are all sigma(k)/3.

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A103288 Numbers k such that sigma(k) >= 2k-1 (union of perfect, abundant and least deficient numbers).

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A285615 Numbers k such that usigma(k) >= 3*k, where usigma(k) = sum of unitary divisors of k (A034448).

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A328135 Exponential 3-abundant numbers: numbers m such that esigma(m) >= 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

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A204828 Numbers n with abundancy 3 <= sigma(n)/n < 4.

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A293187 Bi-unitary 3-abundant numbers: numbers k such that bsigma(k) > 3*k, where bsigma is the sum of the bi-unitary divisors function (A188999).

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