A246033 "Convex" primes: extremal primes in the sense of Tutaj.

2, 3, 7, 19, 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, 155893, 187477, 194119

Offset: 1

Author

Michel Marcus and N. J. A. Sloane, Aug 18 2014

Keywords

• nonn

Comments

See Tutaj (2014) for the precise definition.

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..500
• Nathan McNew, The Most Frequent Values of the Largest Prime Divisor Function, Exper. Math., 2017, Vol. 26, No. 2, 210-224; also arXiv:1504.05985 [math.NT], 2015.
• Carl Pomerance, The Prime Number Graph, Mathematics of Computations, Volume 33, 145, January 1979, pages 399-408.
• Edward Tutaj, Prime numbers with a certain extremal type property, arXiv:1408.3609 [math.NT], 2014.

Crossrefs

A different notion of convex prime is mentioned in A167844.

Programs

Maple

plist := [2,3] ;
nlist := [1,2] ;
p := 5 ;
for n from 3 to 100000 do # experimental upper limit!
    plist := [op(plist),p] ;
    nlist := [op(nlist),n] ;
    doflat := true ;
    while doflat do
        doflat := false ;
        for nrew from nops(nlist)-1 to 2 by -1 do
            slopold := (nlist[nrew]-nlist[nrew-1])/(plist[nrew]-plist[nrew-1]) ;
            slop := (nlist[nrew+1]-nlist[nrew])/(plist[nrew+1]-plist[nrew]) ;
            if slop >= slopold then
                plist := subsop(nrew=NULL,plist) ;
                nlist := subsop(nrew=NULL,nlist) ;
                doflat := true ;
            end if;
        end do:
    end do:
    print(plist) ;
    p := nextprime(p) ;
end do: # R. J. Mathar, Jul 28 2017

Mathematica

terms = 50; nmax0 = 25000; dnmax = 1000; Clear[f];
f[nmax_] := f[nmax] = Module[{}, plist = {2, 3}; nlist = {1, 2}; p = 5;
For[n = 3, n <= nmax, n++,
plist = Append[plist, p];
nlist = Append[nlist, n]; doflat = True;
While[doflat, doflat = False;
For[nrew = Length[nlist]-1, nrew >= 2, nrew--, slopold = (nlist[[nrew]] - nlist[[nrew-1]])/(plist[[nrew]] - plist[[nrew-1]]); slop = (nlist[[nrew+1]] - nlist[[nrew]])/(plist[[nrew+1]] - plist[[nrew]]); If [slop >= slopold, plist [[nrew]] = Nothing nlist[[nrew]] = Nothing; doflat = True]]
]; p = NextPrime[p]
]; PadRight[plist, terms]
];
f[nmax = nmax0]; f[nmax = nmax + dnmax];
While[Print[nmax]; f[nmax][[1 ;; terms]] != f[nmax - dnmax][[1 ;; terms]], nmax = nmax + dnmax];
f[nmax] (* Jean-François Alcover, Nov 01 2018, from R. J. Mathar's Maple code *)

Extensions

a(14) corrected by Edward Tutaj and Charles R Greathouse IV, Nov 27 2014

Primes beyond 33647 from R. J. Mathar, Jul 28 2017