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A197638
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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.
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6
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4, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526
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OFFSET
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1,1
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COMMENTS
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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".
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LINKS
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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.
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EXAMPLE
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4 is a member because G(4) > 0 > G(2) = G(4/2).
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MAPLE
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See "Computation of GA1 numbers".
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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