A088982 - OEIS

%I #39 Aug 29 2024 10:49:58

%S 7,13,19,23,29,37,43,47,53,61,71,73,79,89,97,101,103,107,113,131,137,

%T 139,149,151,163,167,173,181,193,197,199,223,227,229,233,239,251,257,

%U 263,269,271,281,293,307,311,313,317,337,347,349,359,373,379,383,389

%N Primes that are between consecutive prime-indexed primes.

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

%C 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

%H Harvey P. Dale, <a href="/A088982/b088982.txt">Table of n, a(n) for n = 1..5000</a>

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

%F a(n) = prime(composite(n)) = A000040(A002808(n)). - _Terry D. Grant_, Aug 16 2016

%e 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.

%t Flatten[Table[Prime[Range[Prime[n]+1,Prime[n+1]-1]],{n,30}]] (* _Harvey P. Dale_, Mar 22 2015 *)

%o (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",")); ); ) }

%o (PARI) forcomposite(n=2,100,print1(prime(n)",")) \\ _M. F. Hasler_, Jul 31 2015

%Y Essentially the same as A007821.

%K nonn

%O 1,1

%A _Cino Hilliard_, Oct 31 2003