

A005101


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


180



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 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 nth 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)


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
Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), 137143.
Walter Nissen, Abundancy : Some Resources
P. Pollack, C. Pomerance, Some problems of Erdos on the sumofdivisors 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


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
A001065(a(n)) > a(n).  Reinhard Zumkeller, Nov 01 2015


MAPLE

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


MATHEMATICA

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 *)


PROG

(PARI) isA005101(n) = (sigma(n) > 2*n) \\ Michael B. Porter, Nov 07 2009
(Haskell)
a005101 n = a005101_list !! (n1)
a005101_list = filter (\x > a001065 x > x) [1..]
 Reinhard Zumkeller, Nov 01 2015, 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).
Cf. A001065.
Cf. A000396 (perfect numbers).
Sequence in context: A182854 A270660 A173490 * A124626 A231547 A087245
Adjacent sequences: A005098 A005099 A005100 * A005102 A005103 A005104


KEYWORD

nonn,easy,core,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



