1,2

Also n such that sigma_{-1}(n) > sigma_{-1}(m) for all m < n, where sigma_{-1}(n) is the sum of the reciprocals of the divisors of n. - Matthew Vandermast (ghodges14(AT)comcast.net), Jun 09 2004

Alaoglu and Erdos show that: (1) n is superabundant => n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2>=e_3>=...>=e_p (and e_p is 1 unless n=4 or n=36); (2) if q<r are primes, then | e_r - floor(e_q*log(q)/log(r)) | <= 1; (3) q^{e_q}<2^{e_2+2} for primes q, 2<q<=p. - Keith Briggs (keith.briggs(AT)bt.com), Apr 26 2005

L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.

A. Akbary and Z. Friggstad, Superabundant numbers and the Riemann hypothesis, Amer. Math. Monthly, 116 (2009), 273-275.

R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 112.

J. Sandor, "Abundant numbers", In: M. Hazewinkel, Encyclopedia of Mathematics, Supplement III, Kluwer Acad.Publ., 2002 (see pp. 19-21).

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.

D. Kilminster, Table of n, a(n) for n=0..2000 (Extends to n=8436 in the comments.)

Matthew M. Conroy, Home page (listed instead of email address)

P. Erdos & J.-L. Nicolas, Repartition des nombres superabondants (Text in French)

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.

Walter Nissen, Abundancy : Some Resources

T. D. Noe, First 500 superabundant numbers

T. D. Noe, First 1000000 superabundant numbers (21 MB, zipped) [From T. D. Noe (noe(AT)sspectra.com), Oct 15 2009]

Eric Weisstein's World of Mathematics, Superabundant Number

Wikipedia, Superabundant number

a=0; Do[b=DivisorSigma[1, n]/n; If[b>a, a=b; Print[n]], {n, 1, 10^7}]

Cf. A002182, A002093; colossally abundant numbers: A004490.

A023199 is a subsequence. Almost same as A077006.

Cf. A112974 (number of superabundant numbers between colossally abundant numbers).

Sequence in context: A094348 A002182 A077006 this_sequence A166981 A137425 A141320

Adjacent sequences: A004391 A004392 A004393 this_sequence A004395 A004396 A004397

nonn,nice

Matthew Conroy

Matthew Conroy points out that these are different from the highly composite numbers - see A002182. Jul 10 1996.