

A005101


Abundant numbers (sum of divisors of n exceeds 2n).
(Formerly M4825)


157



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
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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), 264296.
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 !! (n1)
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


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from David W. Wilson


STATUS

approved



