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A005101
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Abundant numbers (sum of divisors of n exceeds 2n).
(Formerly M4825)
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120
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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; refs; listen; history; internal format)
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OFFSET
| 1,1
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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 232th abundant number!
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]
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.
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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).
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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
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FORMULA
| a(n) is asymptotic to C*n with C=4.038.. (Deleglise 1998) - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 04 2002
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MAPLE
| 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
| 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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PROG
| (PARI) isA005101(n) = (sigma(n) > 2*n) [From Michael Porter (michael_b_porter(AT)yahoo.com), Nov 07 2009]
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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 A087245 A153501
Adjacent sequences: A005098 A005099 A005100 * A005102 A005103 A005104
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KEYWORD
| nonn,easy,core,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from David W. Wilson (davidwwilson(AT)comcast.net).
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