What is a VP Number™?

This puzzle is based off the What is a Word™ and What is a Phrase™ series started by JLee and the likewise inspired What is a Number™ series.

If a number conforms to a certain rule, I call it a VP number and if not, I call it a non-VP number. Use the following examples to find the rule:

$$\begin{array}{|c|c|} \hline \textbf{VP numbers™}&\textbf{Not VP numbers™}\\ \hline \text{3}&\text{5}\\ \hline \text{547}&\text{520}\\ \hline \text{1297}&\text{1292}\\ \hline \text{2027}&\text{2016}\\ \hline \text{2749}&\text{2741}\\ \hline \text{3593}&\text{3481}\\ \hline \text{4421}&\text{4400}\\ \hline \text{5281}&\text{5099}\\ \hline \text{6217}&\text{6211}\\ \hline \end{array}$$

For those who want, here is a CSV version

VP Number™,Not VP Number™
3,5
547,520
1297,1292
2027,2016
2749,2741
3593,3481
4421,4400
5281,5099
6217,6211


The puzzle relies on the series' inbuilt assumption, that each number can be tested for whether it is a VP Number™ or a Non VP Number™ on its own. In particular, a number's relationship to other numbers in the sequence is irrelevant.

These are not the only VP-Numbers. More of them exist and can be found.

• Whoever downvoted, any specific reason? – Sid Nov 14 '16 at 16:11
• If I manage to find a valid rule that isn't the same one as the one in your mind, is my answer correct? – Buffer Over Read Nov 28 '16 at 0:31
• @CipherRiddle let's see what you have got. – Sid Nov 28 '16 at 2:28
• There may be infinitely many valid answers to be pedantic. :D – Buffer Over Read Dec 11 '16 at 2:37

All the VP numbers

are the set of prime numbers whose positions in the sequence of primes are themselves the first primes in successive prime centuries (OEIS A157338). (—thanks OEIS, now I know what to call these!)

The first 9 values in that series are 2,101,211,307,401,503,601,701,809.
The 2nd prime number is 3.
The 101st prime is 547.
The 211th prime is 1297.
The 307th prime is 2027.
The 401st prime is 2749.
The 503th prime is 3593.
The 601st prime is 4421.
The 701st prime is 5281.
The 809th prime is 6217.

Thus, the first 9 VP numbers are those listed in order in the puzzle.

I originally described this as ...

all Nth prime numbers for each N that is the first value in (k×100..k×100+99) that is prime.
In other words, for each set of one hundred numbers (0-99, 100-199, ...), find the first value in that set that is a prime number, and call it N; for each such N, the Nth prime number is a VP Number.

The first 9 VP numbers (i.e. k in 0..8) are listed in the puzzle:
- 3 is the 2nd prime number. 2 is the first number between 0 and 99 that is prime.
- 547 is the 101st prime number. 101 is the first number between 100 and 199 that is prime.
- 1297 is the 211th prime number. 211 is the first number between 200 and 299 that is prime.
and so on.

They are called "VP Numbers" because they are

(shamelessly stolen from @lois6b's comment. I couldn't resist.)

• wow, very Prime, much nice – lois6b Nov 14 '16 at 16:42
• Rubio you have 3333 rep, 3 silver badges and 33 bronze :P – Beastly Gerbil Nov 14 '16 at 17:13
• @BeastlyGerbil I apologize for the +1 – David Starkey Nov 14 '16 at 18:01
• I'd like to nominate the joint effort by @lois6b and Rubio for puzzle solution of the year. – feelinferrety Jun 20 '17 at 14:09
• wooooww @feelinferrety thanks.but I only added the meme – lois6b Jun 21 '17 at 7:42