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%I M0514 #83 Jan 15 2022 15:34:25
%S 1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,31,32,
%T 33,34,35,37,38,39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,61,62,
%U 63,64,65,67,68,69,71,73,74,75,76,77,79,81,82,83,85,86
%N Deficient numbers: numbers k such that sigma(k) < 2k.
%C A number k is abundant if sigma(k) > 2k (cf. A005101), perfect if sigma(k) = 2k (cf. A000396), or deficient if sigma(k) < 2k (this sequence), where sigma(k) is the sum of the divisors of k (A000203).
%C Also, numbers k such that A033630(k) = 1. - _Reinhard Zumkeller_, Mar 02 2007
%C According to Deléglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480. Since the perfect numbers have density 0, the deficient numbers have density 0.7520 < 1 - A(2) < 0.7526. Thus the n-th deficient number is asymptotic to 1.3287*n < n/(1 - A(2)) < 1.3298*n. - _Daniel Forgues_, Oct 10 2015
%C The data begins with 3 runs of 5 consecutive terms, from 1 to 5, 7 to 11 and 13 to 17. The maximal length of a run of consecutive terms is 5 because 6 is a perfect number and its proper multiples are abundant numbers. - _Bernard Schott_, May 19 2019
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
%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="/A005100/b005100.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 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 Jose Arnaldo Bebita Dris, <a href="https://arxiv.org/abs/1308.6767">A Criterion for Deficient Numbers Using the Abundancy Index and Deficiency Functions</a>, arXiv:1308.6767 [math.NT], 2013-2016; Journal for Algebra and Number Theory Academia, Volume 8, Issue 1 (February 2018), 1-9.
%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/DeficientNumber.html">Deficient 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/Deficient_number">Deficient number</a>.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>.
%F A001065(a(n)) < a(n). - _Reinhard Zumkeller_, Oct 31 2015
%p with(numtheory); s := proc(n) local i,j,ans; ans := [ ]; j := 0; for i while j<n do if sigma(i)<2*i then ans := [ op(ans),i ]; j := j+1; fi; od; RETURN(ans); end; # s(k) returns terms of sequence through k
%p isA005100 := proc(n)
%p numtheory[sigma](n) < 2*n ;
%p end proc:
%p A005100 := proc(n)
%p option remember;
%p local a;
%p if n = 1 then
%p 1;
%p else
%p for a from procname(n-1)+1 do
%p if isA005100(a) then
%p return a;
%p end if;
%p end do:
%p end if;
%p end proc: # _R. J. Mathar_, Jul 08 2015
%t Select[Range[100], DivisorSigma[1, # ] < 2*# &] (* _Stefan Steinerberger_, Mar 31 2006 *)
%o (PARI) isA005100(n) = (sigma(n) < 2*n) \\ _Michael B. Porter_, Nov 08 2009
%o (PARI) for(n=1, 100, if(sigma(n) < 2*n, print1(n", "))) \\ _Altug Alkan_, Oct 15 2015
%o (Haskell)
%o a005100 n = a005100_list !! (n-1)
%o a005100_list = filter (\x -> a001065 x < x) [1..]
%o -- _Reinhard Zumkeller_, Oct 31 2015
%o (Python)
%o from sympy import divisors
%o def ok(n): return sum(divisors(n)) < 2*n
%o print(list(filter(ok, range(1, 87)))) # _Michael S. Branicky_, Aug 29 2021
%o (Python)
%o from sympy import divisor_sigma
%o from itertools import count, islice
%o def A005100_gen(startvalue=1): return filter(lambda n:divisor_sigma(n) < 2*n,count(max(startvalue,1))) # generator of terms >= startvalue
%o A005100_list = list(islice(A005100_gen(),20)) # _Chai Wah Wu_, Jan 14 2022
%Y Cf. A005101 (abundant), A125499 (even deficient), A247328 (odd deficient), A023196 (complement).
%Y By definition, the weird numbers A006037 are not in this sequence.
%Y Cf. A001065, A318172.
%K nonn,easy,core,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Stefan Steinerberger_, Mar 31 2006
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