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Given the sequence of integer numbers. Needed to find rule which was used to produce these numbers.

The sequence is :

0, 0, 1, 2, 5, 6, 9, 10, 13, 18, 19, 24, 27, 28, 31, 36.

Here are first 16 members of the sequence. It can be infinitely extended if necessary.

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    The OEIS finds [A065890](https://oeis.org/A065890). – 2017-01-24
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    http://math.stackexchange.com/a/1790666/44121 – 2017-01-24
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    Can you explain where you found this sequence? (I would remove my vote to close...) – 2017-01-25
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    This sequence is the number of composite numbers which are less than next prime integer. For instance if we consider the sequence of primes: 2, 3, 5, 7, 11, ... than the number of composite numbers which precede them will be: 0, 0, 1, 2, 5, etc. – 2017-01-25
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    Since the definition involves prime numbers, a simple, straightforward formula is unlikely. It's important to note, though, that if you didn't know the definition, many possible alternate definitions are possible. For instance, you can easily come up with a sequence of the first $n$ primes followed by a composite number, and it has a mathematical formula. e.g., http://oeis.org/A059999 – 2017-01-25

2 Answers 2

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As commented before, there are infinite solutions. We only need to consider the family of Lagrange interpolation polynomials $\mathcal{F}_{\lambda}=\{P_{\lambda}(x):\lambda\in\mathbb{R}\}$ satysfying the conditions $$P_{\lambda}(i)=y_i,\;(i=1,2,\ldots 16)\qquad P_{\lambda}(17)=\lambda,$$ where $y_1=0,y_2=0,y_3=1,\ldots,y_{16}=36 $ are the given numbers.

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    probably this answer should be updated wrt problem edits. – 2017-01-25
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    @ioann.voronov: Done! – 2017-01-25
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try http://oeis.org/

There you will find two ansers