%I #14 Dec 11 2014 10:25:03

%S 12,6,24,36,18,30,60,8,4,48,20,72,120,84,16,42,10,40,180,90,96,144,

%T 240,168,108,360,28,54,420,252,132,80,216,210,32,126,300,336,480,56,

%U 192,288,720,840,66,504,156,540,150,264,14,600,140,270,1260,432,78,1080

%N 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.

%C 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).

%C Lim_n->infinity alpha(n) = 0.

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

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

%H Jaroslav Krizek, <a href="/A234516/b234516.txt">Table of n, a(n) for n = 1..200</a>

%e 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)).

%o (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

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

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Jan 03 2014