A112342 Number of primes between (n-th composite - 1)^2 and (n-th composite)^2.

2, 2, 3, 4, 3, 4, 5, 4, 6, 5, 6, 7, 7, 6, 9, 8, 7, 8, 8, 10, 9, 10, 9, 10, 9, 12, 11, 11, 12, 11, 13, 13, 15, 10, 11, 15, 12, 13, 11, 12, 17, 16, 13, 17, 15, 14, 16, 15, 17, 13, 15, 17, 17, 18, 22, 14, 23, 13, 20, 20, 17, 16, 21, 22, 18, 20, 20, 19, 23, 21, 21, 22, 23, 21, 22, 21, 21

Offset

1,1

Comments

Declare a current total to 0 and a current composite to 4 and a current index n to 1. For each prime p, compute c:=floor(sqrt(p)) + 1. If c is composite then see if c is equal to the current composite. If it is, add 1 to the current total. Else a(n) = current total and set the current total to 1. If enough terms of the sequence have been found, store the found terms and end the algorithm. - David A. Corneth, Jun 03 2024

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

David A. Corneth, PARI program

Formula

a(n) = A000720(A002808(n)^2) - A000720((A002808(n)-1)^2). - Ray Chandler, Sep 06 2005

Example

a(1)=2 because for primes 11 and 13, the floor of the square root of both primes is 3. Since 1 is added to each, 3+1=4, for the composite 4 the count is 2.

Mathematica

Table[PrimePi[ResourceFunction["Composite"][n]^2]-PrimePi[(ResourceFunction["Composite"][n]-1)^2], {n, 77}] (* James C. McMahon, Jun 02 2024 *)

Prog

(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
(PARI) \\ See Corneth link

Crossrefs

Cf. A000720, A002808, A112341.

Sequence in context: A328803 A328804 A064514 * A256094 A063712 A340458

Adjacent sequences: A112339 A112340 A112341 * A112343 A112344 A112345

Keyword

nonn,easy

Author

Enoch Haga, Sep 05 2005

Extensions

Edited by Ray Chandler, Sep 06 2005

Status

approved