A234516 Composite numbers n sorted by decreasing values of alpha(n) = log_n(sigma(n)) - log_n(n+1), where sigma(n) = A000203(n) = the sum of divisors of n.

12, 6, 24, 36, 18, 30, 60, 8, 4, 48, 20, 72, 120, 84, 16, 42, 10, 40, 180, 90, 96, 144, 240, 168, 108, 360, 28, 54, 420, 252, 132, 80, 216, 210, 32, 126, 300, 336, 480, 56, 192, 288, 720, 840, 66, 504, 156, 540, 150, 264, 14, 600, 140, 270, 1260, 432, 78, 1080

Offset: 1

Author

Jaroslav Krizek, Jan 03 2014

Keywords

• nonn

Comments

The number alpha(n) = log_n(sigma(n)) - log_n(n+1) = log_n[sigma(n) / (n+1)] is called the alpha-deviation from primality of number n; alpha(p) = 0 for p = prime. See A234520 for definition of beta(n).

Lim_n->infinity alpha(n) = 0.

Conjecture: Every composite number n has a unique value of alpha(n).

Conjecture: sequence A234517 is not the sequence of numbers from a(n) such that a(n) > a(k) for all k < n.

Examples

			For the number 12; alpha(12) = log_12(sigma(12)) - log_12(12+1) = log_12(28) - log_12(13) = 0.308766187… = A234518 (maximal value of function alpha(n)).
		

Links

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

Crossrefs

Cf. A234515, A234517, A234518, A234519, A234520, A234521, A234522, A234523, A234524.

Programs

PARI

lista(nn) = {v = vector(nn, n, if ((n==1) || isprime(n), 0, log(sigma(n)/(n+1))/log(n))); v = vecsort(v,,5); for (i=1, 80, print1(v[i], ", "));} \\ Michel Marcus, Dec 10 2014