Density of Primes

It's well known that the density of primes decreases as we proceed along the number line but, in the range of numbers up to 100,000, where can we find intervals where the density of primes is quite high. To quantify this density, let's take a prime and consider the next FIVE primes that follow it. Now let's calculate the difference between this sixth prime and the first and call this difference the "gap". Thus we have primes 1 to 6 and the gap is given by: $$\textbf{gap = prime 6 - prime 1}$$ Where is this gap equal to 14 (which is minimum possible)? We'll identify the position by reference to the first prime and the gap will tell us the sixth prime because: $$\textbf{prime 6 = prime 1 + gap}$$ And so we have the following gap statistics: $$\textbf{gaps of 14 occur at }\\ 3, 5$$ $$\textbf{gaps of 16 occur at }\\7, 97, 16057, 19417, 43777$$ $$\textbf{gaps of 18 occur at} \\11, 13, 29, 223, 1289,\\ 1481, 1861, 4783,5639, 5641, 13679,\\ 27733, 44263, 80669, 88799, 88801, 93479$$ $$\textbf{gaps of 20 occur at}\\17, 23, 41, 53, 59, 89, \\179, 263, 599, 641, 809, 1277, \\1283, 1601, 1607, 3449, 3527, 3911, \\4001, 4637, 5849, 9419, 14543, 18041, \\19421, 21011, 22271, 26681, 26711, 43781, \\45119,51419, 54401, 55331, 62969, 65699, \\71327, 75983, 87539, 88793, 97367, 97841$$ Figure 1 shows a plot of the various primes (up to 100,000) and their associated gaps. The largest gap of 154 occurs at 69499 and thus the interval is from 69499 to 69653.

Figure 1: permalink

What I've considered is just one measure of prime density. The decision to consider the gap between six successive primes is quite arbitrary. I could have considered five or seven.