A380136 Positions of high water marks for the numbers of primes between prime(n)^2 and prime(n+1)^2.

1, 2, 3, 4, 6, 8, 9, 11, 15, 18, 21, 23, 24, 30, 42, 46, 47, 61, 62, 66, 91, 97, 99, 137, 146, 150, 154, 180, 189, 217, 327, 367, 429, 462, 574, 590, 650, 708, 738, 842, 890, 928, 985, 1006, 1051, 1059, 1183, 1409, 1457, 1532, 1663, 1831, 2191, 2225, 2810

Offset: 1

Author

Eric W. Weisstein, Jan 13 2025

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Brocard's Conjecture.

Crossrefs

Cf. A380135 (High water marks for the numbers).

Cf. A050216 (Number of primes between (prime(n))^2 and (prime(n+1))^2).

Programs

Maple

A050216 := proc(n)
    option remember ;
    local p,pn ;
    if n = 0 then
        2;
    else
        p := ithprime(n) ;
        pn := nextprime(p) ;
        numtheory[pi](pn^2)-numtheory[pi](p^2) ;
    end if;
end proc:
A380136 := proc(n)
    option remember ;
    if n = 1 then
        1;
    else
        for a from  procname(n-1)+1 do
            if A050216(a) > A050216(procname(n-1)) then
                return a ;
            end if;
        end do:
    end if;
end proc:
seq(A380136(n),n=1..30) ; # R. J. Mathar, Jan 27 2025