A247328 -id:A247328 - cp's OEIS frontend

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

N. J. A. Sloane

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

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.

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.

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.

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

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

Select[Range[100], DivisorSigma[1, # ] < 2*# &] (* Stefan Steinerberger, Mar 31 2006 *)

isA005100(n) = (sigma(n) < 2*n) \\ Michael B. Porter, Nov 08 2009

for(n=1, 100, if(sigma(n) < 2*n, print1(n", "))) \\  Altug Alkan, Oct 15 2015

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

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

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

More terms from Stefan Steinerberger, Mar 31 2006

A187793 Sum of the deficient divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 6, 8, 15, 13, 18, 12, 10, 14, 24, 24, 31, 18, 15, 20, 22, 32, 36, 24, 18, 31, 42, 40, 28, 30, 36, 32, 63, 48, 54, 48, 19, 38, 60, 56, 30, 42, 48, 44, 84, 78, 72, 48, 34, 57, 93, 72, 98, 54, 42, 72, 36, 80, 90, 60, 40, 62, 96, 104, 127, 84, 72, 68, 126, 96, 74, 72, 27

Offset: 1

Timothy L. Tiffin, Jan 06 2013

Sum of divisors d of n with sigma(d) < 2*d.
a(n) = sigma(n) when n is itself also deficient.
Also, a(n) agrees with the terms in A117553 except when n is a multiple (k > 1) of either a perfect number or a primitive abundant number.
Notice that a(1) = 1. The remaining fixed points are given by A125310. - Timothy L. Tiffin, Jun 23 2016
a(A028982(n)) is an odd integer.  Also, if n is an odd abundant number that is not a perfect square and n has an odd number of abundant divisors (e.g., 945 has one abundant divisor and 4725 has three abundant divisors), then a(n) will also be odd: a(945) = 975 and a(4725) = 2675. - Timothy L. Tiffin, Jul 18 2016

			a(12) = 10 because the divisors of 12 are 1, 2, 3, 4, 6, 12; of these, 1, 2, 3, 4 are deficient, and they add up to 10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors

Cf. A000203, A005100, A028982, A080226, A117553, A125310, A125499, A187794, A187795, A247328, A274338, A274339, A274340, A274380, A274549, A274829, A294886, A294934.

A187793 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if numtheory[sigma](d) < 2*d then
            a := a+d ;
        end if ;
    end do:
    a ;
end proc:# R. J. Mathar, May 08 2019

Table[Total@ Select[Divisors@ n, DivisorSigma[1, #] < 2 # &], {n, 72}] (* Michael De Vlieger, Jul 18 2016 *)

a(n)=sumdiv(n,d,if(sigma(d,-1)<2,d,0)) \\ Charles R Greathouse IV, Jan 07 2013

From Antti Karttunen, Nov 14 2017: (Start)
a(n) = Sum_{d|n} A294934(d)*d.
a(n) = A294886(n) + (A294934(n)*n).
a(n) + A187794(n) + A187795(n) = A000203(n).
(End)

a(54) corrected by Charles R Greathouse IV, Jan 07 2013

A267872 Number of ON (black) cells in the n-th iteration of the "Rule 237" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 1, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125

Offset: 0

Robert Price, Jan 21 2016

a(n) = A247328(n+1) for 2 <= n < 472, but a(472) = 945 differs from A247328(473) = 947. Furthermore, a(n) = A163985(n+1) for 2 <= n <= 1000. - Georg Fischer, Oct 22 2018

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A163985, A267870.

rule=237; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[Total[catri[[k]]],{k,1,rows}] (* Number of Black cells in stage n *)

Conjectures from Colin Barker, Jan 22 2016 and Apr 20 2019: (Start)
a(n) = 2*a(n-1)-a(n-2) for n>3.
G.f.: (1-x+4*x^2-2*x^3) / (1-x)^2.
(End)

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

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A187793 Sum of the deficient divisors of n.

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A267872 Number of ON (black) cells in the n-th iteration of the "Rule 237" elementary cellular automaton starting with a single ON (black) cell.

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