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A005101
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Abundant numbers (sum of divisors of n exceeds 2n).
(Formerly M4825)
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220
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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
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OFFSET
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1,1
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COMMENTS
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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).
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 Deléglise (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)
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REFERENCES
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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
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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 Deléglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), 137-143.
Jason Earls, On Smarandache repunit n numbers, in Smarandache Notions Journal (2004), Vol. 14.1, page 243.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
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
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FORMULA
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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
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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:
isA005101 := proc(n)
simplify(numtheory[sigma](n) > 2*n) ;
end proc: # R. J. Mathar, Jun 18 2015
A005101 := proc(n)
option remember ;
local a ;
if n =1 then
12 ;
else
a := procname(n-1)+1 ;
while numtheory[sigma](a) <= 2*a do
a := a+1 ;
end do ;
a ;
end if ;
end proc: # R. J. Mathar, Oct 11 2017
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MATHEMATICA
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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 *)
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PROG
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(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
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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).
Cf. A173490 (even abundant numbers).
Cf. A001065.
Cf. A000396 (perfect numbers).
Sequence in context: A328930 A270660 A173490 * A124626 A231547 A290141
Adjacent sequences: A005098 A005099 A005100 * A005102 A005103 A005104
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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
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STATUS
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approved
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