A319583 Numbers in base 10 that are palindromic in bases 2, 4, and 16.

0, 1, 3, 5, 15, 17, 51, 85, 255, 257, 273, 771, 819, 1285, 1365, 3855, 4095, 4097, 4369, 12291, 13107, 20485, 21845, 61455, 65535, 65537, 65793, 69649, 69905, 196611, 197379, 208947, 209715, 327685, 328965, 348245, 349525, 983055, 986895, 1044735, 1048575

Offset: 1

Author

Jeremias M. Gomes, Sep 23 2018

Keywords

• nonn
• base
• look

Comments

Intersection of A006995, A014192, and A029730.

This sequence is infinite as it contains 15*(1 + 16^k) for any k > 0. - Rémy Sigrist, Sep 23 2018

Palindromes in base 16 whose nonzero base-16 digits are either all 1, all 3, all 5 or all F (15). - Robert Israel, Nov 12 2023

Examples

			255 is 11111111 in binary, 3333 in quaternary and FF in hexadecimal. Hence 255 is in the sequence.
Although 21 is 10101 in binary and 111 in quaternary, it is 15 in hexadecimal and therefore not in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006995 (base 2), A014192 (base 4), and A029730 (base 16).

Programs

Magma

[n: n in [0..2*10^7] | Intseq(n, 2) eq Reverse(Intseq(n, 2)) and Intseq(n, 4) eq Reverse(Intseq(n, 4)) and Intseq(n, 16) eq Reverse(Intseq(n, 16))]; // Vincenzo Librandi, Sep 24 2018

Maple

extend:= proc(x, d)
  local a,b,m;
  if d::odd then
    m:= (d-1)/2;
    a:= x mod 16^(m+1);
    b:= floor(x/16^m);
    a + 16^(m+1)*b
  else
    m:= d/2;
    a:= x mod 16^m;
    b:= floor(x/16^m);
    (a + 16^(m+1)*b, a + 16^m * (a mod 16) + 16^(m+1)*b)
  fi
end proc:
V:= [1,3,5,15]: R:= 0, op(V):
for d from 1 to 6 do
  V:= map(extend,V,d);
  R:= R, op(V);
od:
R; # Robert Israel, Nov 12 2023

Mathematica

palQ[n_, b_] := PalindromeQ[IntegerDigits[n, b]];
Reap[Do[If[palQ[n, 2] && palQ[n, 4] && palQ[n, 16], Print[n]; Sow[n]], {n, 0, 10^6}]][[2, 1]] (* Jean-François Alcover, Sep 25 2018 *)

Sage

[n for n in (0..1000) if Word(n.digits(2)).is_palindrome() and Word(n.digits(4)).is_palindrome() and Word(n.digits(16)).is_palindrome()]