A088982 Primes that are between consecutive prime-indexed primes.

7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 79, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 151, 163, 167, 173, 181, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 281, 293, 307, 311, 313, 317, 337, 347, 349, 359, 373, 379, 383, 389

Offset: 1

Author

Cino Hilliard, Oct 31 2003

Keywords

• nonn

Comments

Conjecture: For x > 1 there is at least 1 prime between prime(prime(x)) and prime(prime(x+1)).

This conjecture is equivalent to saying that there is at least one prime index between prime(x) and prime(x+1), which is trivially true because both are odd for x > 1; one has prime(prime(x)) < prime(prime(x)+1) < prime(prime(x+1)). Obviously the definition is equivalent to "primes > 2 with nonprime index", i.e., sequence A007821 without the initial 2. - M. F. Hasler, Jul 31 2015

Examples

			Prime(prime(4)) = 17 and prime(prime(5)) = 31 and 19,23,29 are between 17 and 31, so 19, 23 and 29 are members.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..5000

Crossrefs

Essentially the same as A007821.

Programs

Mathematica

Flatten[Table[Prime[Range[Prime[n]+1,Prime[n+1]-1]],{n,30}]] (* Harvey P. Dale, Mar 22 2015 *)

PARI

pipprimes(n) = { for(x=1,n, c=-2; p1 = prime(prime(x)); p2 = prime(prime(x+1)); forprime(y=p1,p2,c++; if(y > p1 && y < p2,print1(y",")); ); ) }

PARI

forcomposite(n=2,100,print1(prime(n)",")) \\ M. F. Hasler, Jul 31 2015

Formula

Primes p such that prime(prime(x)) < p < prime(prime(x+1)).

a(n) = prime(composite(n)) = A000040(A002808(n)). - Terry D. Grant, Aug 16 2016