Primes between consecutive prime-index-primes

[Henry in Rotherhithe]

It looks rather simple.

1) For x>1, prime(x) is odd.
2) There is a even number (e.g. y = prime(x)+1) between prime(x) and
prime(x+1) for x>1.
3) There is a prime number (namely prime(y)) between prime(prime(x)) and
prime(prime(x+1)) for x>1.

> -----Original Message-----
> From: cino hilliard
> Sent: 31 October 2003 22:15
> To: seqfan at ext.jussieu.fr
> Subject: Primes between consecutive prime-index-primes
>
>
> Hi seq fans,
> I submitted a sequence of primes that are between prime(prime(x)) and
> prime(prime(x+1).
>
> This is similar to  A007821
>            2,7,13,19,23,29,37,43,47,53,61,71,73,79,89,97,101,103,107,
>            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
>            Primes p(n) where n is composite.
>
> Except 2 is not allowed in my sequence.
>
> Conjecture: For x > 1 there is at least 1 prime between prime(prime(x) and
> prime(prime(x+1).
>
> Can someone prove this or provide a link?
>
> Thank,
> Cino
>
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