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A005101 Abundant numbers (sum of divisors of n exceeds 2n).
(Formerly M4825)
180
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 n is abundant if sigma(n) > 2n (this entry), perfect if sigma(n) = 2n (cf. A000396), deficient if sigma(n) < 2n (cf. A005100), where sigma(n) is the sum of the divisors of n (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 n abundant and > 23, the result of (2*A001055)-A101113 is NOT 0. - Eric Desbiaux, Jun 01 2009

If n is a member so is every positive multiple of n. "Primitive" members are in A091191.

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

According to Del├ęglise (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., 44 (1913), 264-296.

R. K. Guy, Unsolved Problems in Number Theory, B2.

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 Del├ęglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), 137-143.

Walter Nissen, Abundancy : Some Resources

P. Pollack, C. Pomerance, Some problems of Erdos on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, 2015, to appear.

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... (Deleglise 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

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

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).

Sequence in context: A182854 A270660 A173490 * A124626 A231547 A087245

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 May 29 16:42 EDT 2017. Contains 287250 sequences.