A263206 - OEIS

%I #8 May 26 2020 14:39:46

%S 2,2,2,2,2,2,2,2,2,2,2,3,3,2,3,2,2,3,3,3,2,3,5,3,1,3,3,3,3,3,5,4,4,3,

%T 1,3,5,5,4,4,5,4,1,4,4,2,5,5,3,4,6,5,4,4,4,5,5,5,4,3,4,4,5,5,5,6,5,5,

%U 6,5,4,4,5,6,6,4,4,7,5,5,7,4,4,5,5,6,6,5,6,7,6,7,7,5,5,5,5,7,7,4

%N Number of primes p with n^2 < prime(p) < (n+2)^2.

%C Conjecture: a(n) > 0 for all n > 0. In other words, for each n = 1,2,3,... the interval (n^2, (n+2)^2) contains a prime with prime subscript.

%C We also guess that a(n) = 1 only for n = 25, 35, 43.

%H Zhi-Wei Sun, <a href="/A263206/b263206.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1) = 2 since 1^2 < prime(2) = 3 < prime(3) = 5 < (1+2)^2 with 2 and 3 both prime.

%e a(25) = 1 since 25^2 = 625 < prime(127) = 709 < (25+2)^2 = 729 with 127 prime.

%e a(35) = 1 since 35^2 = 1225 < prime(211) = 1297 < (35+2)^2 = 1369 with 211 prime.

%e a(43) = 1 since 43^2 = 1849 < prime(293) = 1913 < (43+2)^2 = 2025 with 293 prime.

%t Do[r=0; Do[If[PrimeQ[k], r=r+1], {k, PrimePi[n^2]+1, PrimePi[(n+2)^2-1]}]; Print[n, " ", r]; Continue, {n, 1, 100}]

%t Table[Count[PrimePi/@Select[Range[n^2+1,(n+2)^2-1],PrimeQ],_?PrimeQ],{n,100}] (* _Harvey P. Dale_, May 26 2020 *)

%Y Cf. A000040, A000290, A006450, A263204.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Oct 12 2015