A391413 Last of 3 consecutive primes p1 < p2 < p3 such that the pattern of differences [p2-p1, p3-p2] does not occur earlier.

5, 7, 11, 13, 29, 31, 37, 41, 59, 97, 101, 127, 131, 149, 151, 211, 223, 227, 257, 307, 337, 347, 373, 409, 419, 479, 487, 521, 523, 541, 547, 641, 709, 797, 809, 907, 911, 967, 1087, 1091, 1151, 1153, 1213, 1277, 1279, 1361, 1367, 1399, 1409, 1549, 1657, 1663, 1693, 1697

Offset

1,1

Links

James C. McMahon, Table of n, a(n) for n = 1..1000

Karl-Heinz Hofmann, Table of n, a(n), A391411(n) and A391412(n) for n = 1..20000, with pairs of differences.

Formula

a(n) = A151800(A391412(n)). - Alois P. Heinz, Dec 10 2025

Example

See A391411.

Mathematica

p3={}; s={}; Do[p=Prime[i]; d1=NextPrime[p]-p; d2=NextPrime[p, 2]-NextPrime[p]; d={d1, d2}; If[!MemberQ[s, d], AppendTo[s, d]; AppendTo[p3, NextPrime[p, 2]]], {i, 263}]; p3 (* James C. McMahon, Dec 09 2025 *)

Prog

(PARI) \\ See A391411 with replacement p1 -> p3 in print1.
(Python)
from sympy import nextprime
p1, p2, p3, occurlist, A391413 = 2, 3, 5, [[1, 2]], [5]
while len(A391413) < 54:
    p1 = p2; p2 = p3; p3 = nextprime(p3)
    if (pair:=[p2-p1, p3-p2]) not in occurlist: A391413.append(p3); occurlist.append(pair)
print(A391413) # Karl-Heinz Hofmann, Dec 10 2025

Crossrefs

Cf. A151800, A391411, A391412.

Sequence in context: A200143 A265780 A135774 * A180946 A265006 A135930

Adjacent sequences: A391410 A391411 A391412 * A391414 A391415 A391416

Keyword

nonn

Author

Hugo Pfoertner, Dec 09 2025

Status

approved