A354474 Numbers that can be written as reversals in two different bases, where the bases are also reversals of each other. (Trailing zeros are not allowed.)

65, 67, 75, 85, 96, 130, 134, 150, 170, 192, 195, 225, 255, 288, 300, 327, 340, 375, 381, 425, 433, 450, 487, 510, 525, 595, 600, 654, 665, 667, 675, 680, 750, 762, 765, 795, 825, 895, 900, 927, 974, 975, 981, 996, 1050, 1125, 1200, 1275, 1277, 1308, 1330, 1334, 1340, 1350, 1535, 1590

Offset

1,1

Comments

Proved infinite by Álvaro Lozano-Robledo.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..2000 (terms 1..300 from Robert Israel)

Jordan Canevari, Python program

Jordan Canevari, Data for each value of sequence

Á. Lozano-Robledo and Jordan Canevari, Proof that the sequence is infinite

Example

65 =  (4,1)_16  = (1,4)_61.
134 = (10,4)_13 = (4,10)_31.

Maple

revb:= proc(n, b) local L, i;
  L:= convert(n, base, b);
  add(L[-i]*b^(i-1), i=1..nops(L))
end proc:
filter:= proc(n) local b1, b2, L1, L2;
  for b1 from 12 to n-2 do
    b2:= revb(b1, 10);
    if b2 <= b1 then next fi;
    L1:= convert(n, base, b1);
    L2:= convert(n, base, b2);
    if L1 = ListTools:-Reverse(L2) then return true fi
 od;
 false
end proc:
select(filter, [$1..2000]); # Robert Israel, Feb 03 2026

Prog

(Python) # See Canevari link.
(Python)
from sympy.ntheory import digits
def ok(n): return any(digits(n, b)[1:] == digits(n, int(sr))[1:][::-1] for b in range(12, n-1) if (sb:=str(b))[-1] != "0" and (sr:=sb[::-1]) > sb)
print([k for k in range(1600) if ok(k)]) # Michael S. Branicky, Feb 03 2026

Crossrefs

Subsequence of A355313.

Sequence in context: A373274 A174928 A355313 * A081647 A257444 A045025

Adjacent sequences: A354471 A354472 A354473 * A354475 A354476 A354477

Keyword

nonn,base

Author

Jordan Canevari, Jun 25 2022

Status

approved