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%I #10 Apr 15 2018 14:16:16
%S 4,5,6,7,8,10,11,12,17,18,24,28,29,37,53,59,67,79,89,95,97,121,123,
%T 125,223,305,329,479,521,539,541,905,1087,1147,1277,1345,1351,1355,
%U 1357,5779,8923,10003,11773,12883,19371,19651,19657,28277,31445
%N Numbers k at which the ratio A288814(k) / k reaches a record high.
%e A288814(10) / 10 = 21 / 10 and for n < 10 all ratios are less than 21 / 10, so 10 is in the sequence.
%t Block[{s = Array[If[PrimeQ@ #, 0, Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #]] &, 10^5, 2], t}, t = Table[(1 + FirstPosition[s, k][[1]])/k, {k, 4, LengthWhile[Differences@ Rest@ Union@ s, # == 1 &]}]; Map[3 + FirstPosition[t, #][[1]] &, Union@ FoldList[Max, t]]] (* _Michael De Vlieger_, Mar 26 2018 *)
%o (PARI) isok(k, n) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]) == n;
%o f(n) = forcomposite(k=1, , if (isok(k, n), return(k))); /* A288814 */
%o lista(nn) = {maxr = 0; for (n=4, nn, if ((newr=f(n)/n) > maxr, print1(n, ", "); maxr = newr););} \\ _Michel Marcus_, Mar 26 2018
%Y Cf. A288814.
%K nonn
%O 1,1
%A _David A. Corneth_, Mar 24 2018
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