A341088 a(n) is the least prime p such that there is a set of n primes <= p such that every concatenation of two distinct members of the set is prime.

2, 7, 67, 673, 8389

Offset

5,1

Comments

a(6) > 400000 if it exists.

Consider the graph whose vertices are primes, with an edge {p,q} if both concatenations pq and qp are prime. a(n) is the least p such that there is an n-clique in this graph with largest vertex p.

Links

Table of n, a(n) for n=5..9.

Example

For n=1 there are no concatenations to consider, so a(1) is the first prime 2.
a(2) = 7 with the set {3,7}, where both concatenations 37 and 73 are both prime.
a(3) = 67 with the set {3, 37, 67}, where all concatenations 337, 373, 367, 673, 3767 and 6737 are prime.
a(4) = 673 with the set {3, 7, 109, 673} and 12 prime concatenations.
a(5) = 8389 with the set {13, 5197, 5701, 6733, 8389} and 20 prime concatenations.

Maple

dcat:= proc(x, y) 10^(1+ilog10(y))*x+y end proc:
Primes:= select(isprime, [seq(i, i=3..10^5, 2)]):
nP:= nops(Primes):
V:= Vector(5):
V[1]:= 2: count:= 1:
for i from 1 while count < 5 do
  p:= Primes[i];
  S[p]:= select(t -> isprime(dcat(p, t)) and isprime(dcat(t, p)), Primes[1..i-1]);
  E:= map(convert, select(t -> member(t[1], S[t[2]]), {seq(seq([S[p][i], S[p][j]], i=1..j-1), j=1..nops(S[p]))}), set);
  G:= GraphTheory:-Graph(S[p], E);
  c:= GraphTheory:-CliqueNumber(G)+1;
  if V[c] = 0 then V[c]:= p; count:= count+1 fi
od:
convert(V, list);

Crossrefs

Sequence in context: A246865 A133237 A099660 * A207978 A307246 A225156

Adjacent sequences: A341085 A341086 A341087 * A341089 A341090 A341091

Keyword

nonn,base,more

Author

J. M. Bergot and Robert Israel, Feb 13 2022

Status

approved