OFFSET
1,1
COMMENTS
The members with exactly two prime divisors counted with multiplicity are 4 and 2*p, for primes p > 5. (See Section 5 of "Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis".)
The smallest member with more than two prime factors is 183783600. Such GA1 numbers are called "proper" - see A201557 and "Table of proper GA1 numbers up to 10^60".
The smallest odd member is 1058462574572984015114271643676625.
See "On SA, CA, and GA numbers".
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), article A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384 and arXiv:1112.6010 [math.NT], 2011-2012.
J.-L. Nicolas, Computation of GA1 numbers, 2011.
J.-L. Nicolas, Table of proper GA1 numbers up to 10^60, 2011.
EXAMPLE
4 is a member because G(4) > 0 > G(2) = G(4/2).
MAPLE
See "Computation of GA1 numbers".
MATHEMATICA
g[k_] := g[k] = DivisorSigma[1, k]/(k*Log[Log[k]]); okQ[n_] := Module[{p = Transpose[FactorInteger[n]][[1]]}, i = 1; While[i <= Length[p] && g[n] >= g[n/p[[i]]], i++]; i > Length[p]]; Select[Range[2, 1000], ! PrimeQ[#] && okQ[#] &] (* T. D. Noe, Dec 03 2011 *)
PROG
(PARI) g(k) = sigma(k)/(k*log(log(k)));
isga1(k)=if (isprime(k), return (0)); gk = g(k); f = factor(k); for(i=1, length(f~), if (gk < g(k/f[i, 1]), return(0)); ); 1; \\ Michel Marcus, Sep 09 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Caveney, Jean-Louis Nicolas and Jonathan Sondow, Dec 02 2011
STATUS
approved