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

%I A319583 #38 Nov 12 2023 12:22:01
%S A319583 0,1,3,5,15,17,51,85,255,257,273,771,819,1285,1365,3855,4095,4097,
%T A319583 4369,12291,13107,20485,21845,61455,65535,65537,65793,69649,69905,
%U A319583 196611,197379,208947,209715,327685,328965,348245,349525,983055,986895,1044735,1048575
%N A319583 Numbers in base 10 that are palindromic in bases 2, 4, and 16.
%C A319583 Intersection of A006995, A014192, and A029730.
%C A319583 This sequence is infinite as it contains 15*(1 + 16^k) for any k > 0. - _Rémy Sigrist_, Sep 23 2018
%C A319583 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
%H A319583 Robert Israel, <a href="/A319583/b319583.txt">Table of n, a(n) for n = 1..10000</a>
%e A319583 255 is 11111111 in binary, 3333 in quaternary and FF in hexadecimal. Hence 255 is in the sequence.
%e A319583 Although 21 is 10101 in binary and 111 in quaternary, it is 15 in hexadecimal and therefore not in the sequence.
%p A319583 extend:= proc(x, d)
%p A319583   local a,b,m;
%p A319583   if d::odd then
%p A319583     m:= (d-1)/2;
%p A319583     a:= x mod 16^(m+1);
%p A319583     b:= floor(x/16^m);
%p A319583     a + 16^(m+1)*b
%p A319583   else
%p A319583     m:= d/2;
%p A319583     a:= x mod 16^m;
%p A319583     b:= floor(x/16^m);
%p A319583     (a + 16^(m+1)*b, a + 16^m * (a mod 16) + 16^(m+1)*b)
%p A319583   fi
%p A319583 end proc:
%p A319583 V:= [1,3,5,15]: R:= 0, op(V):
%p A319583 for d from 1 to 6 do
%p A319583   V:= map(extend,V,d);
%p A319583   R:= R, op(V);
%p A319583 od:
%p A319583 R; # _Robert Israel_, Nov 12 2023
%t A319583 palQ[n_, b_] := PalindromeQ[IntegerDigits[n, b]];
%t A319583 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 *)
%o A319583 (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()]
%o A319583 (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
%Y A319583 Cf. A006995 (base 2), A014192 (base 4), and A029730 (base 16).
%K A319583 nonn,base,look
%O A319583 1,3
%A A319583 _Jeremias M. Gomes_, Sep 23 2018