A372142 - OEIS

A372142

a(n) is the smallest prime p such that there exist exactly n distinct primes q where q < p and the representation of p in base q is a palindrome.

2, 3, 31, 443, 23053, 86677, 11827763, 27362989, 755827199, 1306369439

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OFFSET

0,1

COMMENTS

This is a special case of A372141.

It need not be the case that a(n) is a palindrome in base 2, as 23053 is a counterexample.

For p > 3, one only needs to check q such that q^2 + 1 <= p else p = cc_q = c*(q+1), not prime for c != 1 and q != 2. A similar argument shows that p cannot have an even number of digits in base q, else it would be divisible by (q+1). - Michael S. Branicky, Apr 21 2024

LINKS

Table of n, a(n) for n=0..9.

EXAMPLE

a(5) = 86677, as it is palindromic in base 2, 107, 113, 151, and 233, and no smaller number satisfies the property.

PROG

(Python)

from math import isqrt

from sympy import sieve

from sympy.ntheory import digits

from itertools import islice

def ispal(v): return v == v[::-1]

def f(p): return sum(1 for q in sieve.primerange(1, isqrt(p-1)+1) if ispal(digits(p, q)[1:]))