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

2, 3, 5, 7, 13, 19, 23, 31, 43, 47, 73, 113, 199, 283, 467, 661, 887, 1129, 1327, 1627, 2803, 3947, 4297, 5881, 6379, 7043, 9949, 10343, 13187, 15823, 18461, 24137, 33647, 34763, 37663, 42863, 43067, 59753, 59797, 82619, 96017, 102679, 129643, 130699, 142237

Offset

1,1

Comments

"Convex hull of PrimePi" is a short wording for "the upper convex hull of the points {p, PrimePi(p)} for p >= 2".

Links

Table of n, a(n) for n=1..45.

Wikipedia, Convex hull

Example

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).

Mathematica

terms = 42;
pMax = 110000;
a[1] = 2;
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];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 42}]

Prog

(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

Crossrefs

Cf. A000720, A124661, A167844, A246033 (a subsequence).

Sequence in context: A385504 A124661 A385503 * A134266 A233043 A231099

Adjacent sequences: A319123 A319124 A319125 * A319127 A319128 A319129

Keyword

nonn

Author

Jean-François Alcover, Sep 11 2018

Extensions

More terms from Jinyuan Wang, Feb 25 2025

Status

approved