cp's OEIS Frontend

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

%I A174023 #13 Nov 05 2022 22:06:36
%S A174023 2,3,6,9,17,18,20,28,25,30,41,46,41,53,56,73,62,66,81,93,85,84,89,97,
%T A174023 101,127,121,122,119,128,150,141,144,152,150,143,174,203,197,195,196,
%U A174023 194,213,213,218,223,230,235,249,258,256,244,264,262,274,275,278,295
%N A174023 The number of primes between prime(n)# and prime(n)# + prime(n)^2.
%C A174023 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.
%H A174023 T. D. Noe, <a href="/A174023/b174023.txt">Table of n, a(n) for n = 1..100</a>
%F A174023 Limit_{N->infinity} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = 1. - _Alain Rocchelli_, Nov 03 2022
%e A174023 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.
%t A174023 Table[p=Prime[n]; prod=prod*p; Length[Select[Range[prod+1,prod+p^2-1], PrimeQ]], {n,50}]
%K A174023 nonn
%O A174023 1,1
%A A174023 _T. D. Noe_, Mar 12 2010