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A005100 Deficient numbers: numbers k such that sigma(k) < 2k.
(Formerly M0514)
200
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, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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).
Also, numbers k such that A033630(k) = 1. - Reinhard Zumkeller, Mar 02 2007
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
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
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), pp. 137-143.
Jose Arnaldo Bebita Dris, A Criterion for Deficient Numbers Using the Abundancy Index and Deficiency Functions, arXiv:1308.6767 [math.NT], 2013-2016; Journal for Algebra and Number Theory Academia, Volume 8, Issue 1 (February 2018), 1-9.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
Eric Weisstein's World of Mathematics, Deficient Number.
Eric Weisstein's World of Mathematics, Abundance.
Wikipedia, Deficient number.
FORMULA
A001065(a(n)) < a(n). - Reinhard Zumkeller, Oct 31 2015
MAPLE
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
isA005100 := proc(n)
numtheory[sigma](n) < 2*n ;
end proc:
A005100 := proc(n)
option remember;
local a;
if n = 1 then
1;
else
for a from procname(n-1)+1 do
if isA005100(a) then
return a;
end if;
end do:
end if;
end proc: # R. J. Mathar, Jul 08 2015
MATHEMATICA
Select[Range[100], DivisorSigma[1, # ] < 2*# &] (* Stefan Steinerberger, Mar 31 2006 *)
PROG
(PARI) isA005100(n) = (sigma(n) < 2*n) \\ Michael B. Porter, Nov 08 2009
(PARI) for(n=1, 100, if(sigma(n) < 2*n, print1(n", "))) \\ Altug Alkan, Oct 15 2015
(Haskell)
a005100 n = a005100_list !! (n-1)
a005100_list = filter (\x -> a001065 x < x) [1..]
-- Reinhard Zumkeller, Oct 31 2015
(Python)
from sympy import divisors
def ok(n): return sum(divisors(n)) < 2*n
print(list(filter(ok, range(1, 87)))) # Michael S. Branicky, Aug 29 2021
(Python)
from sympy import divisor_sigma
from itertools import count, islice
def A005100_gen(startvalue=1): return filter(lambda n:divisor_sigma(n) < 2*n, count(max(startvalue, 1))) # generator of terms >= startvalue
A005100_list = list(islice(A005100_gen(), 20)) # Chai Wah Wu, Jan 14 2022
CROSSREFS
Cf. A005101 (abundant), A125499 (even deficient), A247328 (odd deficient), A023196 (complement).
By definition, the weird numbers A006037 are not in this sequence.
Sequence in context: A088725 A094520 A136447 * A051772 A049093 A098901
KEYWORD
nonn,easy,core,nice
AUTHOR
EXTENSIONS
More terms from Stefan Steinerberger, Mar 31 2006
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)