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

%I A380136 #13 Jan 27 2025 04:42:01
%S A380136 1,2,3,4,6,8,9,11,15,18,21,23,24,30,42,46,47,61,62,66,91,97,99,137,
%T A380136 146,150,154,180,189,217,327,367,429,462,574,590,650,708,738,842,890,
%U A380136 928,985,1006,1051,1059,1183,1409,1457,1532,1663,1831,2191,2225,2810
%N A380136 Positions of high water marks for the numbers of primes between prime(n)^2 and prime(n+1)^2.
%H A380136 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BrocardsConjecture.html">Brocard's Conjecture</a>.
%p A380136 A050216 := proc(n)
%p A380136     option remember ;
%p A380136     local p,pn ;
%p A380136     if n = 0 then
%p A380136         2;
%p A380136     else
%p A380136         p := ithprime(n) ;
%p A380136         pn := nextprime(p) ;
%p A380136         numtheory[pi](pn^2)-numtheory[pi](p^2) ;
%p A380136     end if;
%p A380136 end proc:
%p A380136 A380136 := proc(n)
%p A380136     option remember ;
%p A380136     if n = 1 then
%p A380136         1;
%p A380136     else
%p A380136         for a from  procname(n-1)+1 do
%p A380136             if A050216(a) > A050216(procname(n-1)) then
%p A380136                 return a ;
%p A380136             end if;
%p A380136         end do:
%p A380136     end if;
%p A380136 end proc:
%p A380136 seq(A380136(n),n=1..30) ; # _R. J. Mathar_, Jan 27 2025
%Y A380136 Cf. A380135 (High water marks for the numbers).
%Y A380136 Cf. A050216 (Number of primes between (prime(n))^2 and (prime(n+1))^2).
%K A380136 nonn
%O A380136 1,2
%A A380136 _Eric W. Weisstein_, Jan 13 2025