Teaser 1986: Protracted calculation

From The Sunday Times, 8th October 2000 [link]

In the circle illustrated the numbers represent the sizes of the sectors:

By combining adjacent ones it is possible to find sectors in this circle of sizes 1, 2, 3, … all the way up to 13 (13 being the whole circle). For example:

7 = 4 + 1 + 2.

In a similar fashion, given a circle divided into five sectors in a particular way, it is possible to combine adjacent sectors to give sizes 1, 2, 3, … up to the biggest possible in the circumstances.

What, in increasing order, are the sizes of the five sectors?

The text of this puzzle is taken from the book Brainteasers (2002, edited by Victor Bryant), the wording differs only slightly from the puzzle originally published in the newspaper.

[teaser1986]