A284798 -id:A284798 - cp's OEIS frontend

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

Stanislav Kruml, May 12 2024

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

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

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.

Cf. A284798.

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

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

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

A284797 Write in base k, complement, reverse. Case k = 3.

Original entry on oeis.org

2, 1, 0, 7, 4, 1, 6, 3, 0, 25, 16, 7, 22, 13, 4, 19, 10, 1, 24, 15, 6, 21, 12, 3, 18, 9, 0, 79, 52, 25, 70, 43, 16, 61, 34, 7, 76, 49, 22, 67, 40, 13, 58, 31, 4, 73, 46, 19, 64, 37, 10, 55, 28, 1, 78, 51, 24, 69, 42, 15, 60, 33, 6, 75, 48, 21, 66, 39, 12, 57, 30

Offset: 0

Paolo P. Lava, Apr 03 2017

			a(9) = 25 because 9 in base 3 is 100, its complement in base 3 is 122 and the digit reverse is 221 that is 25 in base 10.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A036044, A267193, A284798.

P:=proc(q,h) local a,b,k,n; print(h-1); for n from 1 to q do a:=convert(n,base,h); b:=0;
for k from 1 to nops(a) do a[k]:=h-1-a[k]; b:=h*b+a[k]; od; print(b); od; end: P(10^2,3);

Table[FromDigits[Reverse[2-IntegerDigits[n,3]],3],{n,0,70}] (* Harvey P. Dale, Sep 08 2019 *)

from gmpy2 import digits
def A284797(n): return -int((s:=digits(n,3)[::-1]),3)-1+3**len(s) # Chai Wah Wu, Feb 04 2022

cp's OEIS Frontend

A372770 Primes in A284798.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python