A167844 -id:A167844 - cp's OEIS frontend

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

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

nonn

See Tutaj (2014) for the precise definition.

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.

A different notion of convex prime is mentioned in A167844.

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

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 *)

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

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

A167843 Obtuse-angled primes.

Original entry on oeis.org

113, 127, 137, 139, 149, 157, 167, 179, 199, 211, 223, 227, 229, 233, 239, 257, 269, 277, 311, 331, 337, 347, 349, 359, 367, 379, 389, 421, 431, 433, 443, 449, 457, 467, 479, 499, 521, 541, 557, 569, 577, 599, 631, 641, 643, 653, 661, 677, 733, 743, 751, 761

Offset: 1

Omar E. Pol, Nov 13 2009

Primes in A135603.

Primes whose structure of digits represents an obtuse angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right.

Michael S. Branicky, Table of n, a(n) for n = 1..914

Cf. A134811, A134951, A134971, A135603, A167841, A167842, A167844, A167845, A167846, A167853.

# uses agen() from A135603
from sympy import isprime
g = filter(isprime, agen())
print([next(g) for n in range(1, 53)]) # Michael S. Branicky, Aug 03 2022

a(12) and beyond from Michael S. Branicky, Aug 03 2022

A167845 Concave primes.

Original entry on oeis.org

131, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 233, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 331, 353, 367, 373, 379, 383, 389, 397, 431, 443, 461, 463, 467, 479, 487, 491, 499, 541, 563, 571, 577, 587, 593, 599, 641, 653, 661, 673

Offset: 1

Omar E. Pol, Nov 13 2009

Primes in A135642.

Primes whose structure of digits represents a concave function or a concave object. In the graphic representation the points are connected by imaginary line segments from left to right.

Cf. A134811, A134951, A134971, A135642, A167841, A167842, A167843, A167844, A167846, A167853.

More terms from Max Alekseyev, Apr 24 2010

A167846 Concave-convex primes.

Original entry on oeis.org

1021, 1031, 1033, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423

Offset: 1

Omar E. Pol, Nov 13 2009

Primes in A163278.

Prime numbers with more than three digits that are not straight-line numbers (A135643), concave numbers (A135642) or convex numbers (A135641).

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A134811, A134951, A134971, A135641, A135642, A135643, A163278, A167841, A167842, A167843, A167844, A167845, A167853.

More terms from Rémy Sigrist, May 22 2019

A167847 Straight-line primes.

Original entry on oeis.org

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 4567, 76543, 23456789, 1111111111111111111, 11111111111111111111111

Offset: 1

Omar E. Pol, Nov 14 2009

Prime numbers with 2 digits together with the primes whose digits are in arithmetic progression. The structure of digits represents a straight line.
Note that in the graphic representation the points are connected by imaginary line segments (see also A135643).
Note that all two-digit primes are straight-line primes but this sequence has no three-digit terms.
No further terms between 23456789 and 115507867=prime(6600000). - R. J. Mathar, Dec 04 2009
All terms after 23456789 are repunit primes (A004022) with number of digits: 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... (A004023). - Jens Kruse Andersen, Jul 21 2014

			The number 4567 is straight-line prime:
  . . . .
  . . . .
  . . . 7
  . . 6 .
  . 5 . .
  4 . . .
  . . . .
  . . . .
  . . . .
  . . . .

Cf. A000040, A004022, A004023, A134811, A134951, A134971, A135643, A167841, A167842, A167843, A167844, A167845, A167846, A167853.

2 more terms from R. J. Mathar, Dec 04 2009

a(25)-a(26) from Jens Kruse Andersen, Jul 21 2014

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

Original entry on oeis.org

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

Jean-François Alcover, Sep 11 2018

nonn

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

			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).

Wikipedia, Convex hull

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

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}]

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(m0&&sJinyuan Wang, Feb 25 2025

More terms from Jinyuan Wang, Feb 25 2025

cp's OEIS Frontend

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

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A167843 Obtuse-angled primes.

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A167845 Concave primes.

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A167846 Concave-convex primes.

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A167847 Straight-line primes.

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

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Mathematica

PARI

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