1,1

A number n is abundant if sigma(n) > 2n (this entry), perfect if sigma(n) = 2n (cf. A000396), deficient if sigma(n) < 2n (cf. A005100), where sigma(n) is the sum of the divisors of n (A000203).

It appears that for n>23, the result of (2*A001055)-A101113 is NOT 0 if n=A005101. [From Eric Desbiaux (moongerms(AT)wanadoo.fr), Jun 01 2009]

L. E. Dickson, Theorems and tables on the sum of the divisors of a number, Quart. J. Pure Appl. Math., 44 (1913), 264-296.

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

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 59.

T. D. Noe, Table of n, a(n) for n = 1..10000

J. Britton, Perfect Number Analyser

C. K. Caldwell, The Prime Glossary, abundant number

M. Deleglise, Bounds for the density of abundant integers

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Abundance

Wikipedia, Abundant number

Index entries for "core" sequences

a(n) is asymptotic to C*n with C=4.038.. (Deleglise 1998) - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 04 2002

If n is a member so is every positive multiple of n. "Primitive" members are in A091191.

with(numtheory): for n from 1 to 270 do if sigma(n)>2*n then printf(`%d, `, n) fi: od:

abQ[n_] := DivisorSigma[1, n] > 2n; Select[ Range[270], abQ[ # ] &] (from Robert G. Wilson v (rgwv(at)rgwv.com), Sep 15 2005)

Cf. A005835, A005100, A091194, A091196, A080224, A091191 (primitive).

Cf. A005231 and A006038 (odd abundant numbers).

Cf. A094268 (n consecutive abundant numbers).

Adjacent sequences: A005098 A005099 A005100 this_sequence A005102 A005103 A005104

Sequence in context: A126706 A123711 A059404 this_sequence A124626 A087245 A153501

nonn,easy,core,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from David W. Wilson (davidwwilson(AT)comcast.net).