A174023 The number of primes between prime(n)# and prime(n)# + prime(n)^2.

2, 3, 6, 9, 17, 18, 20, 28, 25, 30, 41, 46, 41, 53, 56, 73, 62, 66, 81, 93, 85, 84, 89, 97, 101, 127, 121, 122, 119, 128, 150, 141, 144, 152, 150, 143, 174, 203, 197, 195, 196, 194, 213, 213, 218, 223, 230, 235, 249, 258, 256, 244, 264, 262, 274, 275, 278, 295

Offset: 1

Author

T. D. Noe, Mar 12 2010

Keywords

• nonn

Comments

Here prime(n)# denotes the product of the first n primes, A002110(n). This sequence provides numerical evidence that the smallest prime p greater than prime(n)#+1 is a prime distance from prime(n)#; that is, p-prime(n)# is a prime number, as shown in the sequence of Fortunate numbers, A005235. For p-prime(n)# to be a composite number, p would have to be greater than prime(n)#+prime(n)^2, which would imply that a(n)=0.

Examples

			For 3, the second prime, 3# is 6 and 3#+3^2 is 15. There are 3 primes between 6 and 15: 7, 11, and 13. Hence a(2)=3.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..100

Programs

Mathematica

Table[p=Prime[n]; prod=prod*p; Length[Select[Range[prod+1,prod+p^2-1], PrimeQ]], {n,50}]

Formula

Limit_{N->infinity} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = 1. - Alain Rocchelli, Nov 03 2022