A358427 - OEIS

%I #8 Nov 19 2022 20:24:40

%S 2,3,13,23,19,353,157,173,101,113,137,193,181,1831,983,1297,2861,1321,

%T 1259,1381,1229,1039,1009,1097,1033,1019,1237,1129,1051,1013,1049,

%U 1723,1181,1117,1583,1523,1153,1439

%N a(n) is the least prime p such that there are exactly n primes q with the same number of digits as p such that the concatenations p|q and q|p are prime, or 0 if there is no such p.

%C For every 5-digit prime p, there are at least 70 primes q.  Thus it is very likely that a(n) = 0 for 38 <= n <= 69.  However, there is no proof of this.

%e a(4) = 19 because 19 is prime and there are 4 primes 13, 31, 79, 97 where 1913, 1319, 1931, 3119, 1979, 7931, 1397 and 9731 are prime, and no smaller prime works.

%p A:= Array(0..37): count:= 0: p:= 0:

%p while count < 38 do

%p   p:= nextprime(p);

%p   v:= f(p);

%p   if v <= 37 and A[v] = 0 then count:= count+1; A[v]:= p; fi;

%p od:

%p convert(A,list);

%o (Python)

%o from itertools import count, islice

%o from sympy import isprime, primerange

%o def agen(): # generator of terms

%o     adict, n = dict(), 0

%o     for d in count(1):

%o         pow = 10**d

%o         for p in primerange(10**(d-1), pow):

%o             v = 0

%o             for q in primerange(10**(d-1), pow):

%o                 t = p*pow + q

%o                 if isprime(p*pow + q) and isprime(q*pow + p): v += 1

%o             if v not in adict: adict[v] = p

%o             while n in adict: yield adict[n]; n += 1

%o print(list(islice(agen(), 38))) # _Michael S. Branicky_, Nov 15 2022

%Y Cf. A358421.

%K nonn,base,more

%O 0,1

%A _J. M. Bergot_ and _Robert Israel_, Nov 15 2022