A174167 Number of safe primes between squares of consecutive primes.

2, 2, 1, 3, 1, 3, 2, 4, 4, 2, 7, 4, 1, 6, 3, 10, 1, 10, 6, 2, 10, 5, 12, 9, 9, 4, 6, 3, 9, 26, 6, 10, 5, 18, 4, 17, 11, 10, 17, 13, 3, 23, 3, 9, 6, 36, 32, 8, 6, 9, 15, 10, 22, 19, 18, 15, 7, 22, 15, 9, 31, 43, 13, 6, 14, 47, 25, 35, 10, 10, 21, 32, 23, 18, 9, 27, 34, 18, 32, 46, 3, 38, 12, 20

Offset: 1

Author

Jaspal Singh Cheema, Mar 10 2010

Keywords

• nonn

Comments

If you graph n vs a(n), interesting patterns begin to emerge. As you go farther along the n-axis, greater are the number of Safe Primes, on average, within each interval obtained. The smallest count of 1 occurs 4 times (terms: 3rd, 5th, 13th, and 17th) in the sequence above. I suspect the number of Safe Primes within each interval will never be zero. If one could prove this, then it would imply that Safe Primes are infinite. Can you prove it?

Examples

			Take any pair of consecutive primes. Let us say the very first one (2,3). Square both terms to obtain an interval (4,9). Within this interval, there are two Safe Primes, namely 5 and 7. Hence the very first term of the sequence above is 2. Similarly, the next term, 2, refers to the two Safe Primes between squares of (3,5), or the interval (9,25), which are 11 and 23.
		

Links

• Jaspal Singh Cheema, Table of n, a(n) for n = 1..10177
• Rick Aster, Prime number sieve, SAS prime sieve program
• Wikipedia, Safe prime

Crossrefs

Cf. A005385, A156875.

Extensions

Edited by D. S. McNeil, Nov 17 2010