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A319126 Convex hull primes, that is, prime numbers corresponding to the convex hull of PrimePi, the prime counting function.

%I A319126 #27 Feb 25 2025 08:59:42
%S A319126 2,3,5,7,13,19,23,31,43,47,73,113,199,283,467,661,887,1129,1327,1627,
%T A319126 2803,3947,4297,5881,6379,7043,9949,10343,13187,15823,18461,24137,
%U A319126 33647,34763,37663,42863,43067,59753,59797,82619,96017,102679,129643,130699,142237
%N A319126 Convex hull primes, that is, prime numbers corresponding to the convex hull of PrimePi, the prime counting function.
%C A319126 "Convex hull of PrimePi" is a short wording for "the upper convex hull of the points {p, PrimePi(p)} for p >= 2".
%H A319126 Wikipedia, <a href="https://en.wikipedia.org/wiki/Convex_hull">Convex hull</a>
%e A319126 Prime 83 is not member because there exist two primes from the convex hull, namely 47 and 113, such that (PrimePi(83) - PrimePi(47))/(83 - 47) < (PrimePi(113) - PrimePi(83))/(113 - 83).
%t A319126 terms = 42;
%t A319126 pMax = 110000;
%t A319126 a[1] = 2;
%t A319126 a[n_] := a[n] = Module[{}, For[slopeMax = 0; p1 = NextPrime[a[n-1]], p1 <= pMax, p1 = NextPrime[p1], slope = (PrimePi[p1] - PrimePi[a[n-1]])/(p1 - a[n-1]); If[slope > slopeMax, slopeMax = slope; p1Max = p1]]; p1Max];
%t A319126 Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 42}]
%o A319126 (PARI) lista(nn) = my(c, m, p=2, r, s, t=1); print1(p); for(n=2, nn, c=t; m=0; forprime(q=p+1, oo, c++; if(m<s=(c-t)/(q-p), m=s; r=q); s=(c-t)/(c*(log(c)+log(log(c))-1)-p); if(s>0&&s<m, break)); print1(", ", r); t=primepi(p=r)); \\ _Jinyuan Wang_, Feb 25 2025
%Y A319126 Cf. A000720, A124661, A167844, A246033 (a subsequence).
%K A319126 nonn
%O A319126 1,1
%A A319126 _Jean-François Alcover_, Sep 11 2018
%E A319126 More terms from _Jinyuan Wang_, Feb 25 2025