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%I #20 Jun 16 2020 05:35:53
%S 1,12,24,36,72,120,144,216,360,432,720,1440,2160,2880,4320,8640,12960,
%T 17280,20160,25920,30240,40320,51840,60480,80640,120960,181440,241920,
%U 362880,483840,604800,725760,967680,1209600,1451520,1814400,2177280,2419200,2903040,3628800
%N Numbers with a record value of the ratio of the number of abundant divisors to the total number of divisors.
%C Apparently, all the terms are least numbers of their prime signature (A025487). This was verified for the first 78 terms.
%C The ratio A080224(m)/A000005(m) can be arbitrarily close to 1. For example, A080224(6^k)/A000005(6^k) = (k-1)/(k+1) = 1 - 2/(k+1), for k >= 1.
%H Amiram Eldar, <a href="/A335541/b335541.txt">Table of n, a(n) for n = 1..78</a>
%F Numbers m such that A080224(m)/A000005(m) > A080224(k)/A000005(k) for all k < m.
%e 36 has 9 divisors, {1, 2, 3, 4, 6, 9, 12, 18, 36}, 3 of which are abundant, {12, 18, 36}. The ratio 3/9 = 1/3 is larger than the ratios for all the numbers below 36. Hence 36 is a term.
%t f[n_] := Count[(d = Divisors[n]), _?(DivisorSigma[1, #] > 2# &)]/Length[d]; fm = -1; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^4}]; s
%Y Cf. A000005, A000203, A005100, A025487, A080224, A335540.
%K nonn
%O 1,2
%A _Amiram Eldar_, Jun 13 2020
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