Primes between consecutive prime-index-primes

[cino hilliard]

Henry in Rotherhithe wrote:

>From: "Henry in Rotherhithe" <se16 at btinternet.com>
>To: "cino hilliard" <hillcino368 at hotmail.com>, <seqfan at ext.jussieu.fr>
>Subject: RE: Primes between consecutive prime-index-primes
>Date: Sat, 1 Nov 2003 00:56:40 -0000
>
>It looks rather simple.

Egads, indeed it was. Thank you and others for the explanation.

>
>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.

I have submitted another sequence that lists the "only" 1  prime between 
prime(prime(x)) and
prime(prime(x+1)).

7,13,37,61,113,181,281,359,557,593,787...

Conjecture: The primes that are the "only" prime between two consecutive
prime-index-primes are infinite.

Also sequence for only 3 primes and trajectory for 1,3,5..2k+1 primes

Conjecture: There cannot be an even number total of primes between two
consecutive prime-index-primes.

Maybe these can also be disposed of easily also.

Cino

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