Games Online Contest #16

5 First Prizes: A one-year subscription to GAMES

This contest closed on February 29, 2008. A new contest will appear soon.
Here was the challenge:

A Trio of Sequences

What integer logically comes next (that is, should replace the question mark) in each of the following sequences (A, B, and C)? Hint: The three integer sequences below are based on three other sequences that are all related—in different ways—to each of these sequences.

A. 4, 9, 18, 30, 47, 65, 88, 112, 141, 175, ?

B. 2, 24, 180, 784, 3025, 7488, 18326, 35264, 68931, 133400, ?

C. 211, 324, 549, 7716, 111125, 131636, 172249, 192964, 233781, 2946100, ?

The correct answers were: A, 208; B, 210056; C , 3156121. The three series are based on three simpler series: the prime numbers, the lazy caterer’s sequence, and the squares (a, b, c, below):

a: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,…
b: 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56,…
c: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121,…

Sequence A’s nth term is obtained by adding up the nth terms in these three series; sequence B’s nth term is the product of the three series’ nth terms; and sequence C’s nth term is simply the concatenation of the nth number strings.

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