A372770 Primes in A284798.

13, 97, 853, 1021, 1093, 7873, 8161, 8377, 9337, 12241, 62989, 63853, 66733, 74797, 79861, 81373, 82021, 84181, 86413, 91381, 92317, 94477, 95773, 98893, 100189, 101701, 111997, 114157, 534841, 552553, 556441, 560977, 578689, 580633, 591937, 600361, 631249

Offset: 1

Author

Stanislav Kruml, May 12 2024

Keywords

• base
• nonn
• easy

Comments

The base-b expansion (d_1)(d_2)...(d_m) of a number is antipalindromic if, for each of its m digits, it holds that d_k + d_{m-k+1} = b-1.

In a base other than 3, there is at most a single antipalindromic prime.

All terms have an odd number of base-3 digits. - Robert Israel, Mar 14 2025

Examples

			For m = 3, the only solution is 13 = 111_3.
For m = 5, the only solution is 97 = 10121_3.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Lubomira Dvorakova, Stanislav Kruml, and David Ryzák, Antipalindromic numbers, arXiv:2008.06864 [math.CO], 2020.

Crossrefs

Cf. A284798.

Programs

Maple

arev3:= proc(n) local L,i;
    L:= convert(n,base,3);
    add((2-L[-i])*3^(i-1),i=1..nops(L))
end proc;
qprime:= proc(x) if isprime(x) then x fi end proc:
F:= proc(d) local x,y;
    seq(qprime(x*3^((d+1)/2) + 3^((d-1)/2) + arev3(x)),x=3^((d-3)/2)..2*3^((d-3)/2)-1)
end proc;
[seq(F(i),i=3..13,2)]; # Robert Israel, Mar 14 2025

Mathematica

Select[Prime[Range[52000]], FromDigits[Reverse[2 - IntegerDigits[#, 3]], 3] == # &] (* Amiram Eldar, Jun 16 2024 *)

Python

from sympy import isprime
from itertools import count, islice, product
def bgen(): # generator of terms of A284798
    yield 1
    for d in count(2):
        for first in [1, 2]:
            for rest in product([0, 1, 2], repeat=(d-2)//2):
                left, mid = (first,) + rest, (1,) if d&1 else tuple()
                right = tuple([2-d for d in left][::-1])
                yield int("".join(str(d) for d in left + mid + right), 3)
def agen(): yield from filter(isprime, bgen())
print(list(islice(agen(), 40))) # Michael S. Branicky, Jun 16 2024