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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• core
• nice

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

If p and q are primes such that phi(p*q) > p+1, then p*q^n is a term in the sequence for all n >= 1 where phi is the Euler totient function. - Amrit Awasthi, Sep 10 2024

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).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 128.

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.
• S. Flora Jeba, Anirban Roy, and Manjil P. Saikia, On k-Facile Perfect Numbers, Algebra and Its Applications (ICAA-2023) Springer Proc. Math. Stat., Vol. 474, 111-121. See p. 112.
• 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.

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.

Programs

Haskell

a005100 n = a005100_list !! (n-1)
a005100_list = filter (\x -> a001065 x < x) [1..]
-- Reinhard Zumkeller, Oct 31 2015

Maple

with(numtheory); s := proc(n) local i,j,ans; ans := [ ]; j := 0; for i while jA005100 := 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 *)

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

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

Formula

A001065(a(n)) < a(n). - Reinhard Zumkeller, Oct 31 2015

Extensions

More terms from Stefan Steinerberger, Mar 31 2006