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%I M4825 #154 Apr 03 2023 10:36:09
%S 12,18,20,24,30,36,40,42,48,54,56,60,66,70,72,78,80,84,88,90,96,100,
%T 102,104,108,112,114,120,126,132,138,140,144,150,156,160,162,168,174,
%U 176,180,186,192,196,198,200,204,208,210,216,220,222,224,228,234,240,246,252,258,260,264,270
%N Abundant numbers (sum of divisors of m exceeds 2m).
%C A number m is abundant if sigma(m) > 2m (this sequence), perfect if sigma(m) = 2m (cf. A000396), or deficient if sigma(m) < 2m (cf. A005100), where sigma(m) is the sum of the divisors of m (A000203).
%C 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!
%C It appears that for m abundant and > 23, 2*A001055(m) - A101113(m) is NOT 0. - _Eric Desbiaux_, Jun 01 2009
%C If m is a term so is every positive multiple of m. "Primitive" terms are in A091191.
%C If m=6k (k>=2), then sigma(m) >= 1 + k + 2*k + 3*k + 6*k > 12*k = 2*m. Thus all such m are in the sequence.
%C 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
%C From _Bob Selcoe_, Mar 28 2017 (prompted by correspondence with Peter Seymour): (Start)
%C Applying similar logic as the proof that all multiples of 6 >= 12 appear in the sequence, for all odd primes p:
%C i) all numbers of the form j*p*2^k (j >= 1) appear in the sequence when p < 2^(k+1) - 1;
%C ii) no numbers appear when p > 2^(k+1) - 1 (i.e., are deficient and are in A005100);
%C iii) when p = 2^(k+1) - 1 (i.e., perfect numbers, A000396), j*p*2^k (j >= 2) appear.
%C Note that redundancies are eliminated when evaluating p only in the interval [2^k, 2^(k+1)].
%C The first few even terms not of the forms i or iii are {70, 350, 490, 550, 572, 650, 770, ...}. (End)
%D L. E. Dickson, Theorems and tables on the sum of the divisors of a number, Quart. J. Pure Appl. Math., Vol. 44 (1913), pp. 264-296.
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 59.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A005101/b005101.txt">Table of n, a(n) for n = 1..10000</a>
%H J. Britton, <a href="http://britton.disted.camosun.bc.ca/perfect/jbperfect.htm">Perfect Number Analyser</a>.
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/AbundantNumber.html">abundant number</a>.
%H Marc Deléglise, <a href="http://projecteuclid.org/euclid.em/1048515661">Bounds for the density of abundant integers</a>, Experiment. Math., Volume 7, Issue 2 (1998), pp. 137-143.
%H Jason Earls, <a href="http://fs.unm.edu/SNJ/OnSmarandacheRepunit.pdf">On Smarandache repunit n numbers</a>, in Smarandache Notions Journal, Vol. 14, No. 1 (2004), page 243.
%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1505.07229v3">The zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, 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.]
%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1610.07793">Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1610.07793 [math.NT], 2016.
%H Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">Abundancy : Some Resources </a>.
%H Paul Pollack and Carl Pomerance, <a href="https://doi.org/10.1090/btran/10">Some problems of Erdős on the sum-of-divisors function</a>, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; <a href="http://pollack.uga.edu/reversal-errata.pdf">Errata</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbundantNumber.html">Abundant Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Abundance.html">Abundance</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Abundant_number">Abundant number</a>.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>.
%F a(n) is asymptotic to C*n with C=4.038... (Deléglise, 1998). - _Benoit Cloitre_, Sep 04 2002
%F A005101 = { n | A033880(n) > 0 }. - _M. F. Hasler_, Apr 19 2012
%F A001065(a(n)) > a(n). - _Reinhard Zumkeller_, Nov 01 2015
%p with(numtheory): for n from 1 to 270 do if sigma(n)>2*n then printf(`%d,`,n) fi: od:
%p isA005101 := proc(n)
%p simplify(numtheory[sigma](n) > 2*n) ;
%p end proc: # _R. J. Mathar_, Jun 18 2015
%p A005101 := proc(n)
%p option remember ;
%p local a ;
%p if n =1 then
%p 12 ;
%p else
%p a := procname(n-1)+1 ;
%p while numtheory[sigma](a) <= 2*a do
%p a := a+1 ;
%p end do ;
%p a ;
%p end if ;
%p end proc: # _R. J. Mathar_, Oct 11 2017
%t abQ[n_] := DivisorSigma[1, n] > 2n; A005101 = Select[ Range[270], abQ[ # ] &] (* _Robert G. Wilson v_, Sep 15 2005 *)
%t Select[Range[300], DivisorSigma[1, #] > 2 # &] (* _Vincenzo Librandi_, Oct 12 2015 *)
%o (PARI) isA005101(n) = (sigma(n) > 2*n) \\ _Michael B. Porter_, Nov 07 2009
%o (Haskell)
%o a005101 n = a005101_list !! (n-1)
%o a005101_list = filter (\x -> a001065 x > x) [1..]
%o -- _Reinhard Zumkeller_, Nov 01 2015, Jan 21 2013
%o (Python)
%o from sympy import divisors
%o def ok(n): return sum(divisors(n)) > 2*n
%o print(list(filter(ok, range(1, 271)))) # _Michael S. Branicky_, Aug 29 2021
%o (Python)
%o from sympy import divisor_sigma
%o from itertools import count, islice
%o def A005101_gen(startvalue=1): return filter(lambda n:divisor_sigma(n) > 2*n, count(max(startvalue, 1))) # generator of terms >= startvalue
%o A005101_list = list(islice(A005101_gen(), 20)) # _Chai Wah Wu_, Jan 14 2022
%Y Cf. A005835, A005100, A091194, A091196, A080224, A091191 (primitive).
%Y Cf. A005231 and A006038 (odd abundant numbers).
%Y Cf. A094268 (n consecutive abundant numbers).
%Y Cf. A173490 (even abundant numbers).
%Y Cf. A001065.
%Y Cf. A000396 (perfect numbers).
%Y Cf. A302991.
%K nonn,easy,core,nice
%O 1,1
%A _N. J. A. Sloane_
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