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A005101 Abundant numbers (sum of divisors of m exceeds 2m).
(Formerly M4825)
294
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

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.

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.

Eric Weisstein's World of Mathematics, Abundant Number.

Eric Weisstein's World of Mathematics, Abundance.

Wikipedia, Abundant number.

Index entries for "core" sequences.

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

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 *)

PROG

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

(Haskell)

a005101 n = a005101_list !! (n-1)

a005101_list = filter (\x -> a001065 x > x) [1..]

-- Reinhard Zumkeller, Nov 01 2015, Jan 21 2013

(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

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.

Sequence in context: A328930 A270660 A173490 * A124626 A231547 A290141

Adjacent sequences:  A005098 A005099 A005100 * A005102 A005103 A005104

KEYWORD

nonn,easy,core,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 24 18:55 EDT 2022. Contains 356949 sequences. (Running on oeis4.)