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.

1, 2, 4, 6, 12, 8, 24, 3, 18, 36, 30, 60, 10, 20, 48, 72, 120, 16, 40, 84, 180, 42, 90, 240, 144, 360, 96, 168, 28, 420, 108, 80, 252, 720, 14, 15, 210, 840, 54, 56, 336, 480, 216, 126, 32, 504, 288, 9, 540, 1260, 300, 132, 140, 1680, 192, 2520, 1080, 600, 630

Offset: 1

Author

Jaroslav Krizek, Jan 18 2014

Keywords

• nonn

Comments

The number k(n) = log_tau(n) (sigma(n)) = log(sigma(n)) / log(tau(n)) is such that tau(n)^k(n) = sigma(n).

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]

See A236021 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

Examples

			For number 1; k(1) = 1.
For number 2; k(2) = log_tau(2) (sigma(2)) = log_2 (3) = 1.5849625007... = A020857.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..1288

Crossrefs

Cf. A000005, A000203, A236021.

Cf. A236022, A236023, A236024, A236025, A236026.

Programs

Mathematica

A[nn_] := Ordering[ N[ Join[ {1}, Table[ Log[DivisorSigma[0, i], DivisorSigma[1, i]], {i, 2, nn} ] ] ] ];
A236020[nn_] := A[nn^2][[1 ;; nn]];
A236020[59] (* Robert P. P. McKone, Jan 17 2021 *)

PARI

\\ warning: does not generate all the terms up to nn
f(k) = if (k==1, 1, log(sigma(k)) / log(numdiv(k)));
lista(nn) = vecsort(vector(nn, k, f(k)),, 1); \\ Michel Marcus, Jan 16 2021