A234520 Composite numbers n sorted by decreasing values of beta(n) = sigma(n)^(1/n) - (n+1)^(1/n), where sigma(n) = A000203(n) = the sum of divisors of n.

4, 6, 8, 12, 10, 18, 16, 24, 14, 20, 9, 15, 30, 36, 28, 22, 32, 40, 48, 42, 21, 26, 60, 54, 44, 27, 72, 56, 34, 50, 45, 52, 38, 66, 84, 33, 64, 90, 80, 70, 96, 78, 46, 39, 120, 68, 108, 35, 88, 76, 63, 25, 100, 58, 102, 126, 144, 112, 132, 62, 104, 75, 51, 92

Offset

1,1

Comments

The number beta(n) = sigma(n)^(1/n) - (n+1)^(1/n) is called the beta-deviation from primality of the number n; beta(p) = 0 for p = prime. See A234516 for definition of alpha(n).

For number 4; beta(4) = sigma(4)^(1/4) - (4+1)^(1/4), = 7^(1/4) - 5^(1/4) = 0,131227780… = A234522 (maximal value of function beta(n)).

Lim_n->infinity beta(n) = 0.

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

See A234523 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

Links

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A234515, A234516, A234517, A234518, A234519, A234521, A234522, A234523, A234524.

Sequence in context: A352374 A110646 A320126 * A275789 A031359 A274790

Adjacent sequences: A234517 A234518 A234519 * A234521 A234522 A234523

Keyword

nonn

Author

Jaroslav Krizek, Jan 14 2014

Status

approved