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

1,1

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.

Links

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

Example

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)).

Prog

(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

Crossrefs

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

Sequence in context: A040135 A004015 A119870 * A177690 A038332 A093763

Adjacent sequences: A234513 A234514 A234515 * A234517 A234518 A234519

Keyword

nonn

Author

Jaroslav Krizek, Jan 03 2014

Status

approved