A234515 Natural numbers n sorted by decreasing values of number k(n) = log_n(sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n.

2, 4, 6, 12, 8, 24, 18, 3, 36, 30, 10, 60, 20, 48, 16, 72, 120, 84, 42, 40, 180, 90, 96, 28, 144, 240, 168, 14, 108, 360, 54, 32, 420, 80, 252, 132, 216, 56, 210, 126, 300, 66, 336, 480, 192, 288, 720, 840, 156, 504, 150, 540, 264, 140, 600, 78, 270, 1260, 432

Offset: 1

Author

Jaroslav Krizek, Jan 03 2014

Keywords

• nonn

Comments

Number k(n) = log_n(sigma(n)) = log(sigma(n)) / log(n) is number such that n^k(n) = sigma(n).

The last term of this infinite sequence is number 1, k(1) = 1 (minimal value of function k(n)).

Conjecture: Every natural number n has a unique value of number k(n).

See A234517 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

Examples

			For number 2; k(2) = log_2(sigma(2)) = log_2(3) = 1,5849625007… = A020857 (maximal value of function k(n)).
		

Links

• Jaroslav Krizek, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A234516, A234517, A234518, A234519, A234520, A234521, A234522, A234523, A234524.

Programs

PARI

lista(nn=100000) = {v = vector(nn, n, if (n==1, 0, log(sigma(n))/log(n))); v = vecsort(v,,5); for (i=1, 80, print1(v[i], ", "));} \\ Michel Marcus, Dec 11 2014