A115941 a(n) is the least prime whose representation contains a palindromic substring of length n.

2, 11, 101, 11113, 10301, 1011013, 1003001, 100110013, 100030001, 10000000019, 10000500001, 1000011000017, 1000008000001, 100000440000011, 100000323000001, 10000001100000011, 10000000500000001, 1000000011000000019, 1000000008000000001, 100000000660000000013

Offset: 1

Author

Giovanni Resta, Feb 06 2006

Keywords

• nonn
• base

Examples

			a(6)=1011013 since it is the least prime that contains a palindromic substring (101101) of length 6.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..25

Crossrefs

Cf. A056732.

Programs

Python

from sympy import isprime
from itertools import product
def pals_to_test(n, odd=True):
  if n <= 2: yield [2, 11][n-1]
  if odd: ruled_out = "024568" # can't be even or multiple of 5
  else: ruled_out = "0"
  midrange = [[""], [str(i) for i in range(10)]]
  for p in product("0123456789", repeat=n//2):
    left = "".join(p)
    if len(left):
      if left[0] in ruled_out: continue
    for middle in midrange[n%2]:
        out = left+middle+left[::-1]
        if odd: yield out
        else:
          for last in "1379": yield out+last
def a(n):
  palsgen = pals_to_test(n, n%2 == 1)
  while True:
    strpal = next(palsgen)
    pal = int(strpal)
    if isprime(pal): return pal
print([a(n) for n in range(1, 18)]) # Michael S. Branicky, Feb 11 2021

Extensions

a(18) and beyond from Michael S. Branicky, Feb 11 2021