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
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).
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.
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
CROSSREFS
By definition, the weird numbers A006037 are not in this sequence.
KEYWORD
nonn,easy,core,nice
AUTHOR
EXTENSIONS
More terms from Stefan Steinerberger, Mar 31 2006
STATUS
approved