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Search: id:A005101
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| A005101 |
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Abundant numbers (sum of divisors of n exceeds 2n).
(Formerly M4825)
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+0
89
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| 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; listen)
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OFFSET
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1,1
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COMMENT
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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).
It appears that for n>23, the result of (2*A001055)-A101113 is NOT 0 if n=A005101. [From Eric Desbiaux (moongerms(AT)wanadoo.fr), Jun 01 2009]
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
Walter Nissen, Abundancy : Some Resources
J. Britton, Perfect Number Analyser
C. K. Caldwell, The Prime Glossary, abundant number
M. Deleglise, Bounds for the density of abundant integers
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Abundance
Wikipedia, Abundant number
Index entries for "core" sequences
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FORMULA
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a(n) is asymptotic to C*n with C=4.038.. (Deleglise 1998) - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 04 2002
If n is a member so is every positive multiple of n. "Primitive" members are in A091191.
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MAPLE
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with(numtheory): for n from 1 to 270 do if sigma(n)>2*n then printf(`%d, `, n) fi: od:
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MATHEMATICA
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abQ[n_] := DivisorSigma[1, n] > 2n; Select[ Range[270], abQ[ # ] &] (from Robert G. Wilson v (rgwv(at)rgwv.com), Sep 15 2005)
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CROSSREFS
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Cf. A005835, A005100, A091194, A091196, A080224, A091191 (primitive).
Cf. A005231 and A006038 (odd abundant numbers).
Cf. A094268 (n consecutive abundant numbers).
Adjacent sequences: A005098 A005099 A005100 this_sequence A005102 A005103 A005104
Sequence in context: A126706 A123711 A059404 this_sequence A124626 A087245 A153501
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KEYWORD
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nonn,easy,core,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from David W. Wilson (davidwwilson(AT)comcast.net).
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