%I #14 Apr 29 2024 09:08:54

%S 2,2,3,4,5,5,7,6,7,9,8,9,12,9,10,16,13,16,15,21,15,18,19,18,21,23,20,

%T 24,23,25,29,28,23,27,33,32,27,32,33,30,29,36,34,37,37,37,38,41,45,38,

%U 39,49,47,45,53,46,53,46,45,49,53,51,48,49,55,51,62,66,61,61,60,66,63,61

%N Number of primes between (prime(n)-1)^2 and prime(n)^2.

%C Begin with the first prime, compute square root, take floor and add 1. If result is a prime number then begin the count for that prime value. Increment the count until prime value changes.

%H Amiram Eldar, <a href="/A112341/b112341.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000720(A000040(n)^2) - A000720((A000040(n)-1)^2). - _Ray Chandler_, Sep 06 2005

%e a(5) = 5 because for primes 101-103-107-109-113 the floor of the square root of each is 10. For each 10, 1 is added, so for prime 11 the count is 5.

%t a[n_] := PrimePi[Prime[n]^2] - PrimePi[(Prime[n] - 1)^2]; Table[a[n], {n, 74}] (* _Ray Chandler_, Sep 06 2005 *)

%o (UBASIC) 10 A=1 20 B=nxtprm(B) 30 C=int(sqrt(B)) 40 D=C+1 50 if E=D then N=N+1:else print N:N=1:stop 60 if D=prmdiv(D) then print B;C;D;"-" 70 E=D 80 goto 20

%o (PARI) lista(pmax) = {my(c = 0); forprime(p = 2, pmax, c = 0; forprime(q = (p-1)^2, p^2, c++); print1(c, ", "));} \\ _Amiram Eldar_, Apr 29 2024

%Y Cf. A000040, A000720, A112342.

%K easy,nonn

%O 1,1

%A _Enoch Haga_, Sep 05 2005

%E Edited by _Ray Chandler_, Sep 06 2005