A344936 a(n) is the smallest prime p such that a string s of n zeros can be inserted between all adjacent digits of p simultaneously such that the resulting number is also prime and is also prime for each s of length k with 0 < k < n.

11, 19, 19, 71, 98689, 130049597, 78736136153

Offset: 1

Author

Felix Fröhlich, Jun 03 2021

Keywords

• nonn
• base
• hard
• more

Comments

a(n) is the smallest prime p such that A344937(i) >= n, where i is the index of p in A000040.

Examples

			For n = 5: 98689, 908060809, 9008006008009, 90008000600080009, 900008000060000800009 and 9000008000006000008000009 are all prime. Since 98689 is the smallest prime where strings of zeros of successive lengths up to 5 can be inserted between all adjacent digits such that each resulting number is also prime, a(5) = 98689.
		

Crossrefs

Cf. A000040, A344637, A344937.

Programs

Mathematica

Table[m=1;While[!And@@Table[PrimeQ@FromDigits@Flatten@Riffle[IntegerDigits@Prime@m,{Table[0,k]}],{k,n}],m++];Prime@m,{n,5}] (* Giorgos Kalogeropoulos, Jun 03 2021 *)

PARI

eva(n) = subst(Pol(n), x, 10)
insert_zeros(num, len) = my(d=digits(num), v=[]); for(k=1, #d-1, v=concat(v, concat([d[k]], vector(len)))); v=concat(v, d[#d]); eva(v)
a(n) = forprime(p=10, , for(k=1, n, if(!ispseudoprime(eva(insert_zeros(p, k))), break, if(k==n, return(p)))))

Python

from sympy import isprime, nextprime
def insert_zeros(n, k): return int(("0"*k).join(list(str(n))))
def ok(p, n): return all(isprime(insert_zeros(p, k)) for k in range(1, n+1))
def a(n, startat=11):
  p = startat
  while True:
    if ok(p, n): return p
    p = nextprime(p)
print([a(n) for n in range(1, 6)]) # Michael S. Branicky, Jun 03 2021

Extensions

a(7) from Michael S. Branicky, Jun 11 2021