A005101 Abundant numbers (sum of divisors of m exceeds 2m).

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• core
• nice

Comments

A number m is abundant if sigma(m) > 2m (this sequence), perfect if sigma(m) = 2m (cf. A000396), or deficient if sigma(m) < 2m (cf. A005100), where sigma(m) is the sum of the divisors of m (A000203).

While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number!

It appears that for m abundant and > 23, 2*A001055(m) - A101113(m) is NOT 0. - Eric Desbiaux, Jun 01 2009

If m is a term so is every positive multiple of m. "Primitive" terms are in A091191.

If m=6k (k>=2), then sigma(m) >= 1 + k + 2*k + 3*k + 6*k > 12*k = 2*m. Thus all such m are in the sequence.

According to Deléglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480. Thus the n-th abundant number is asymptotic to 4.0322*n < n/A(2) < 4.0421*n. - Daniel Forgues, Oct 11 2015

From Bob Selcoe, Mar 28 2017 (prompted by correspondence with Peter Seymour): (Start)

Applying similar logic as the proof that all multiples of 6 >= 12 appear in the sequence, for all odd primes p:

i) all numbers of the form j*p*2^k (j >= 1) appear in the sequence when p < 2^(k+1) - 1;

ii) no numbers appear when p > 2^(k+1) - 1 (i.e., are deficient and are in A005100);

iii) when p = 2^(k+1) - 1 (i.e., perfect numbers, A000396), j*p*2^k (j >= 2) appear.

Note that redundancies are eliminated when evaluating p only in the interval [2^k, 2^(k+1)].

The first few even terms not of the forms i or iii are {70, 350, 490, 550, 572, 650, 770, ...}. (End)

References

• L. E. Dickson, Theorems and tables on the sum of the divisors of a number, Quart. J. Pure Appl. Math., Vol. 44 (1913), pp. 264-296.
• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
• Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 59.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 128.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• J. Britton, Perfect Number Analyser.
• C. K. Caldwell, The Prime Glossary, abundant number.
• Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math., Volume 7, Issue 2 (1998), pp. 137-143.
• Jason Earls, On Smarandache repunit n numbers, in Smarandache Notions Journal, Vol. 14, No. 1 (2004), page 243.
• S. Flora Jeba, Anirban Roy, and Manjil P. Saikia, On k-Facile Perfect Numbers, Algebra and Its Applications (ICAA-2023) Springer Proc. Math. Stat., Vol. 474, 111-121. See p. 112.
• Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]
• Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
• Walter Nissen, Abundancy : Some Resources .
• Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
• Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 10.
• Eric Weisstein's World of Mathematics, Abundant Number.
• Eric Weisstein's World of Mathematics, Abundance.
• Wikipedia, Abundant number.
• Index entries for "core" sequences.

Crossrefs

Cf. A005835, A005100, A091194, A091196, A080224, A091191 (primitive).

Cf. A005231 and A006038 (odd abundant numbers).

Cf. A094268 (n consecutive abundant numbers).

Cf. A173490 (even abundant numbers).

Cf. A001065.

Cf. A000396 (perfect numbers).

Cf. A302991.

Programs

Haskell

a005101 n = a005101_list !! (n-1)
a005101_list = filter (\x -> a001065 x > x) [1..]
-- Reinhard Zumkeller, Nov 01 2015, Jan 21 2013

Maple

with(numtheory): for n from 1 to 270 do if sigma(n)>2*n then printf(`%d,`,n) fi: od:
isA005101 := proc(n)
    simplify(numtheory[sigma](n) > 2*n) ;
end proc: # R. J. Mathar, Jun 18 2015
A005101 := proc(n)
    option remember ;
    local a ;
    if n =1 then
        12 ;
    else
        a := procname(n-1)+1 ;
        while numtheory[sigma](a) <= 2*a do
            a := a+1 ;
        end do ;
        a ;
    end if ;
end proc: # R. J. Mathar, Oct 11 2017

Mathematica

abQ[n_] := DivisorSigma[1, n] > 2n; A005101 = Select[ Range[270], abQ[ # ] &] (* Robert G. Wilson v, Sep 15 2005 *)
Select[Range[300], DivisorSigma[1, #] > 2 # &] (* Vincenzo Librandi, Oct 12 2015 *)

PARI

isA005101(n) = (sigma(n) > 2*n) \\ Michael B. Porter, Nov 07 2009

Python

from sympy import divisors
def ok(n): return sum(divisors(n)) > 2*n
print(list(filter(ok, range(1, 271)))) # Michael S. Branicky, Aug 29 2021

Python

from sympy import divisor_sigma
from itertools import count, islice
def A005101_gen(startvalue=1): return filter(lambda n:divisor_sigma(n) > 2*n, count(max(startvalue, 1))) # generator of terms >= startvalue
A005101_list = list(islice(A005101_gen(), 20)) # Chai Wah Wu, Jan 14 2022

Formula

a(n) is asymptotic to C*n with C=4.038... (Deléglise, 1998). - Benoit Cloitre, Sep 04 2002

A005101 = { n | A033880(n) > 0 }. - M. F. Hasler, Apr 19 2012

A001065(a(n)) > a(n). - Reinhard Zumkeller, Nov 01 2015