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A066523 Crowded numbers: for any k in the sequence, d(k)/k is larger than d(m)/m for all m > k.

%I A066523 #26 Jan 24 2025 11:59:48
%S A066523 2,4,6,12,24,30,36,48,60,72,84,120,144,180,240,252,360,420,480,504,
%T A066523 540,720,840,900,1008,1080,1260,1440,1680,1800,2520,2640,2880,3360,
%U A066523 3780,3960,5040,5280,5400,5460,5544,6300,7560,7920,10080,10920,12600
%N A066523 Crowded numbers: for any k in the sequence, d(k)/k is larger than d(m)/m for all m > k.
%C A066523 Since d(m) < 2*sqrt(m), we need only test values of m < (2k/d(k))^2.
%C A066523 It was briefly conjectured that this sequence was the same as the highly composite numbers (A002182) larger than 1, but this is false: 30 is crowded but not highly composite and 50400 is highly composite but not crowded. Is every super-abundant number (A004394) crowded?
%C A066523 Additional comments from Roy Maulbogat, Jan 22 2008: (Start)
%C A066523 It can easily be shown that all crowded numbers are even and that there is always a crowded number between N and 2N. This allows us to improve the algorithm as follows:
%C A066523 crowded[k_] := Module[{},
%C A066523 * If[OddQ[k], Return [False]];*
%C A066523 div = DivisorSigma[0,k]/k;
%C A066523 For [ *m=k+2, m<=2k, m+=2*, If[
%C A066523 DivisorSigma [0, m] / m<=div, Return [False]]];True];
%C A066523 numlist = Select[Range[1,10^7],crowded]
%C A066523 On second thought, it might be wise to use Min[2k, stop] as the stopping condition of the loop ("stop" being the variable defined in the original algorithm). (End)
%H A066523 Donovan Johnson, <a href="/A066523/b066523.txt">Table of n, a(n) for n = 1..300</a> (first 129 terms from Roy Maulbogat)
%p A066523 Tau := n -> NumberTheory:-tau(n): t := (n, k) -> k*Tau(n) < n*Tau(k):
%p A066523 isCrowded := proc(k) local n; if k::odd then return false fi;
%p A066523 andmap(j -> t(j, k), [seq(k+2..2*k,2)]) end:
%p A066523 aList := n -> select(isCrowded, [seq(1..n)]): aList(1100);  # _Peter Luschny_, Jan 24 2025
%t A066523 crowded[n_] := Module[{}, stop=(2/(dovern=DivisorSigma[0, n]/n))^2; For[m=n+1, m<stop, m++, If[DivisorSigma[0, m]/m>=dovern, Return[False]]]; True]; Select[Range[1, 13000], crowded]
%Y A066523 Cf. A002182, A000005, A004394, A354768.
%K A066523 nonn
%O A066523 1,1
%A A066523 Roy Maulbogat (maulbogat(AT)gmail.com), Jan 05 2002
%E A066523 Edited by _Dean Hickerson_, Jan 07 2002