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A005100 Deficient numbers: numbers n such that sigma(n) < 2n.
(Formerly M0514)
124
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A number n is abundant if sigma(n) > 2n (cf. A005101), perfect if sigma(n) = 2n (cf. A000396), deficient if sigma(n) < 2n (this entry), where sigma(n) is the sum of the divisors of n (A000203).

Also, numbers n such that A033630(n) = 1. - Reinhard Zumkeller, Mar 02 2007

According to Del├ęglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480. Since the perfect numbers having 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

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, B2.

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 Del├ęglise, Bounds for the density of abundant integers, Experiment. Math. Volume 7, Issue 2 (1998), 137-143.

Walter Nissen, Abundancy : Some Resources

P. Pollack, C. Pomerance, Some problems of Erdos on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, 2015, to appear.

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

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.

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

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Last modified August 27 18:12 EDT 2016. Contains 275912 sequences.