A005100 Deficient numbers: numbers k such that sigma(k) < 2k. (Formerly M0514)

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

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

T. D. Noe, Table of n, a(n) for n = 1..10000

J. Britton, Perfect Number Analyser.

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.

Walter Nissen, Abundancy : Some Resources .

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.

Index entries for "core" sequences.

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.

Cf. A001065, A318172.

Sequence in context: A088725 A094520 A136447 * A051772 A049093 A098901

Adjacent sequences: A005097 A005098 A005099 * A005101 A005102 A005103

Keyword

nonn,easy,core,nice

Author

N. J. A. Sloane

Extensions

More terms from Stefan Steinerberger, Mar 31 2006

Status

approved