cp's OEIS Frontend

A236020 Natural numbers n sorted by increasing values of k(n) = log_tau(n) (sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n and tau(n) = A000005(n) = the number of divisors of n.

%I A236020 #35 Jan 20 2021 18:38:18
%S A236020 1,2,4,6,12,8,24,3,18,36,30,60,10,20,48,72,120,16,40,84,180,42,90,240,
%T A236020 144,360,96,168,28,420,108,80,252,720,14,15,210,840,54,56,336,480,216,
%U A236020 126,32,504,288,9,540,1260,300,132,140,1680,192,2520,1080,600,630
%N A236020 Natural numbers n sorted by increasing values of k(n) = log_tau(n) (sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n and tau(n) = A000005(n) = the number of divisors of n.
%C A236020 The number k(n) = log_tau(n) (sigma(n)) = log(sigma(n)) / log(tau(n)) is such that tau(n)^k(n) = sigma(n).
%C A236020 Conjecture: every natural number n has a unique value of k(n). [The conjecture is wrong: e.g., k(5) = k(22) = log(6)/log(2). - _Amiram Eldar_, Jan 17 2021]
%C A236020 See A236021 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.
%H A236020 Michel Marcus, <a href="/A236020/b236020.txt">Table of n, a(n) for n = 1..1288</a>
%e A236020 For number 1; k(1) = 1.
%e A236020 For number 2; k(2) = log_tau(2) (sigma(2)) = log_2 (3) = 1.5849625007... = A020857.
%t A236020 A[nn_] := Ordering[ N[ Join[ {1}, Table[ Log[DivisorSigma[0, i], DivisorSigma[1, i]], {i, 2, nn} ] ] ] ];
%t A236020 A236020[nn_] := A[nn^2][[1 ;; nn]];
%t A236020 A236020[59] (* _Robert P. P. McKone_, Jan 17 2021 *)
%o A236020 (PARI) \\ warning: does not generate all the terms up to nn
%o A236020 f(k) = if (k==1, 1, log(sigma(k)) / log(numdiv(k)));
%o A236020 lista(nn) = vecsort(vector(nn, k, f(k)),, 1); \\ _Michel Marcus_, Jan 16 2021
%Y A236020 Cf. A000005, A000203, A236021.
%Y A236020 Cf. A236022, A236023, A236024, A236025, A236026.
%K A236020 nonn
%O A236020 1,2
%A A236020 _Jaroslav Krizek_, Jan 18 2014