A005100 - OEIS (original) (raw)

A005100

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

232

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

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).

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

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.

Eric Weisstein's World of Mathematics, Abundance.

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:

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;

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..]

(Python)

from sympy import divisors

def ok(n): return sum(divisors(n)) < 2*n

(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.