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%I #38 May 10 2020 04:32:33
%S 4,14,22,26,34,38,46,58,62,74,82,86,94,106,118,122,134,142,146,158,
%T 166,178,194,202,206,214,218,226,254,262,274,278,298,302,314,326,334,
%U 346,358,362,382,386,394,398,422,446,454,458,466,478,482,502,514,526
%N GA1 numbers: composite m with G(m) >= G(m/p) for all prime factors p of m, where G(k) = sigma(k)/(k*log(log(k))) and sigma(k) = sum of divisors of k.
%C 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".)
%C 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".
%C The smallest odd member is 1058462574572984015114271643676625.
%C See "On SA, CA, and GA numbers".
%H Amiram Eldar, <a href="/A197638/b197638.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H G. Caveney, J.-L. Nicolas, and J. Sondow, <a href="http://www.integers-ejcnt.org/l33/l33.pdf">Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis</a>, Integers 11 (2011), article A33.
%H G. Caveney, J.-L. Nicolas and J. Sondow, <a href="http://arxiv.org/abs/1112.6010">On SA, CA, and GA numbers</a>, Ramanujan J., 29 (2012), 359-384 and arXiv:1112.6010 [math.NT], 2011-2012.
%H J.-L. Nicolas, <a href="http://math.univ-lyon1.fr/~nicolas/GAnumbers.html">Computation of GA1 numbers</a>, 2011.
%H J.-L. Nicolas, <a href="http://math.univ-lyon1.fr/~nicolas/GA160">Table of proper GA1 numbers up to 10^60</a>, 2011.
%e 4 is a member because G(4) > 0 > G(2) = G(4/2).
%p See "Computation of GA1 numbers".
%t 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 *)
%o (PARI) g(k) = sigma(k)/(k*log(log(k)));
%o 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
%Y Cf. A000203, A197639, A201557, A201558.
%K nonn
%O 1,1
%A Geoffrey Caveney, Jean-Louis Nicolas and _Jonathan Sondow_, Dec 02 2011