Problem 3

Let

0<f(1)<f(2)<f(3)<\ldots

be a sequence with all its terms positive integers. The

n

-th positive integer which doesn't belong to the sequence is

f(f(n))+1

. Find

f(240)

Step 1 of 5: Fix the first term

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In plain words

The gap rule cannot remove 1, so the sequence must start at 1.

f(1)=1

Detailed analysis

Because $f(1)\ge 1$ , we have $f(f(1))\ge 1$ , so the first missing positive integer is at least $2$ . Thus $1$ belongs to the sequence and, being its least positive term, gives $f(1)=1$ .