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A335541
Numbers with a record value of the ratio of the number of abundant divisors to the total number of divisors.
2
1, 12, 24, 36, 72, 120, 144, 216, 360, 432, 720, 1440, 2160, 2880, 4320, 8640, 12960, 17280, 20160, 25920, 30240, 40320, 51840, 60480, 80640, 120960, 181440, 241920, 362880, 483840, 604800, 725760, 967680, 1209600, 1451520, 1814400, 2177280, 2419200, 2903040, 3628800
OFFSET
1,2
COMMENTS
Apparently, all the terms are least numbers of their prime signature (A025487). This was verified for the first 78 terms.
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.
LINKS
FORMULA
Numbers m such that A080224(m)/A000005(m) > A080224(k)/A000005(k) for all k < m.
EXAMPLE
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.
MATHEMATICA
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jun 13 2020
STATUS
approved