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astonishing numbers
A number n whose representation can be decomposed into two parts, a and b, such that n is equal to the sum of the integers from a to b. more

The astonishing numbers up to 1015 :

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 223 values, from 15 to 891406343154063).

n\r 0  1
2111112 2
31573729 3
454525760 4
5888410815 5
6771916801813 6
788341619192126 7
82525282829272932 8
99710112320103778 9
1034506065434489 10
111008149221613111389

A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.