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

1,2

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.

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000005, A000203, A236021.

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

Sequence in context: A064469 A057700 A337182 * A234515 A136103 A182235

Adjacent sequences: A236017 A236018 A236019 * A236021 A236022 A236023

Keyword

nonn

Author

Jaroslav Krizek, Jan 18 2014

Status

approved