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A005101 Abundant numbers (sum of divisors of n exceeds 2n).
(Formerly M4825)
156
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 > 23, the result of (2*A001055)-A101113 is NOT 0 if n=A005101. - 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.

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

M. Deleglise, Bounds for the density of abundant integers

Walter Nissen, Abundancy : Some Resources

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

MAPLE

with(numtheory): for n from 1 to 270 do if sigma(n)>2*n then printf(`%d, `, n) fi: od:

MATHEMATICA

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

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 -> a000203 x > 2 * x) [1..]

-- Reinhard Zumkeller, 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).

Sequence in context: A177425 A182854 A173490 * A124626 A231547 A087245

Adjacent sequences:  A005098 A005099 A005100 * A005102 A005103 A005104

KEYWORD

nonn,easy,core,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified April 18 03:34 EDT 2014. Contains 240688 sequences.