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%I A307870 #12 May 12 2019 08:53:14
%S A307870 1,4,8,16,32,64,128,256,432,576,864,1296,1728,2592,3456,5184,6912,
%T A307870 10368,15552,20736,31104,41472,62208,82944,93312,124416,186624,248832,
%U A307870 373248,497664,746496,995328,1119744,1492992,2239488,2592000,2985984,3888000,5184000,7776000
%N A307870 Numbers k with record values of the ratio d(k)/ud(k) between the number of divisors and the number of unitary divisors.
%C A307870 Numbers k with d(k)/2^omega(k) > d(j)/2^omega(j) for all j < k, where d(k) is the number of divisors of k (A000005), and omega(k) is the number of distinct prime factors of k (A001221), so 2^omega(k) is the number of unitary divisors of k (A034444).
%C A307870 Subsequence of A025487.
%C A307870 The first term that is divisible by the k-th prime is 4, 432, 2592000, 53343360000, 134190022982400000, 35377857659079936000000, 160601747163451186424832000000, 35800939973308629849857487360000000, ...
%C A307870 All the terms are powerful (A001694), since if p is a prime factor of k with multuplicity 1, then k and k/p have the same ratio.
%H A307870 Amiram Eldar, <a href="/A307870/b307870.txt">Table of n, a(n) for n = 1..809</a>
%e A307870 All squarefree numbers k have d(k)/ud(k) = 1. Thus 4, the first nonsquarefree number, has a record value of d(4)/ud(4) = 3/2 and thus it is in the sequence.
%t A307870 r[n_] := DivisorSigma[0, n]/(2^PrimeNu[n]); rm = 0; n = 1; s = {}; Do[r1 = r[n]; If[r1 > rm, rm = r1; AppendTo[s, n]]; n++, {10^7}]; s
%Y A307870 Cf. A000005, A001221, A001694, A025487, A034444, A285906, A307869.
%K A307870 nonn
%O A307870 1,2
%A A307870 _Amiram Eldar_, May 02 2019

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