The Ramanujan Journal

, Volume 29, Issue 1, pp 359–384

On SA, CA, and GA numbers

  • Geoffrey Caveney
  • Jean-Louis Nicolas
  • Jonathan Sondow

DOI: 10.1007/s11139-012-9371-0

Cite this article as:
Caveney, G., Nicolas, JL. & Sondow, J. Ramanujan J (2012) 29: 359. doi:10.1007/s11139-012-9371-0


Gronwall’s function G is defined for n>1 by \(G(n)=\frac{\sigma(n)}{n \log\log n}\) where σ(n) is the sum of the divisors of n. We call an integer N>1 a GA1 number if N is composite and G(N)≥G(N/p) for all prime factors p of N. We say that N is a GA2 number if G(N)≥G(aN) for all multiples aN of N. In (Caveney et al. Integers 11:A33, 2011), we used Robin’s and Gronwall’s theorems on G to prove that the Riemann Hypothesis (RH) is true if and only if 4 is the only number that is both GA1 and GA2. In the present paper, we study GA1 numbers and GA2 numbers separately. We compare them with superabundant (SA) and colossally abundant (CA) numbers (first studied by Ramanujan). We give algorithms for computing GA1 numbers; the smallest one with more than two prime factors is 183783600, while the smallest odd one is 1058462574572984015114271643676625. We find nineteen GA2 numbers ≤5040, and prove that a GA2 number N>5040 exists if and only if RH is false, in which case N is even and >108576.


Algorithm Colossally abundant Gronwall’s theorem Prime factor Riemann Hypothesis Robin’s inequality Sum-of-divisors function Superabundant 

Mathematics Subject Classification (2000)

11M26 11A41 11Y55 

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Geoffrey Caveney
    • 1
  • Jean-Louis Nicolas
    • 2
  • Jonathan Sondow
    • 3
  1. 1.ChicagoUSA
  2. 2.Université de Lyon; CNRS; Université Lyon 1; Institut Camille Jordan, MathématiquesVilleurbanne cedexFrance
  3. 3.New YorkUSA

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