Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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