A372141 Primes p that are palindromic in some prime base q, where q < p.

3, 5, 7, 13, 17, 23, 31, 41, 67, 71, 73, 83, 107, 109, 127, 151, 157, 173, 199, 233, 257, 271, 277, 307, 313, 353, 379, 409, 419, 421, 431, 443, 457, 499, 521, 523, 571, 587, 599, 601, 631, 643, 647, 653, 691, 701, 709, 719, 733, 743, 757, 787, 797, 809, 823, 829, 857, 863, 887

Offset

1,1

Comments

If we remove either constraint of q < p or q being prime, then the sequence would be all prime numbers (A000040).

By definition it is a superset of A016041, and is a proper superset by construction (e.g., 13 is in the sequence).

Some terms have multiple bases that yield palindromic representations, the first being 31 (which is palindromic in both base 2 and base 5). The smallest prime p such that there exist n distinct primes less than p that give palindromic representations of p is A372142(n).

Links

Chai Wah Wu, Table of n, a(n) for n = 1..11371 (terms 1..1000 from Tadayoshi Kamegai)

Example

11 is not in this sequence as its representation in base 2 is 1011, in base 3 is 102, in base 5 is 21, in base 7 is 14, none of which are palindromic.
1483 is in this sequence as its representation in base 37 is 131, which is palindromic.

Mathematica

a={}; For[i=1, i<=155, i++, flag=0; For[j=1, Prime[j] < Prime[i] && flag==0, j++, If[PalindromeQ[IntegerDigits[Prime[i], Prime[j]]], flag=1; AppendTo[a, Prime[i]]]]]; a (* Stefano Spezia, Apr 22 2024 *)

Prog

(Python)
from sympy import sieve
from sympy.ntheory import digits
from itertools import islice
def ispal(v): return v == v[::-1]
def agen(): yield from (p for p in sieve if any(ispal(digits(p, q)[1:]) for q in sieve.primerange(1, p)))
print(list(islice(agen(), 60))) # Michael S. Branicky, Apr 20 2024

Crossrefs

Cf. A372142, A002385, A002113.

Cf. A007500, A016041, A077798.

Sequence in context: A028879 A067567 A104294 * A262100 A171566 A137270

Adjacent sequences: A372138 A372139 A372140 * A372142 A372143 A372144

Keyword

nonn,base

Author

Tadayoshi Kamegai, Apr 20 2024

Status

approved