A007633 Palindromic in bases 3 and 10.

0, 1, 2, 4, 8, 121, 151, 212, 242, 484, 656, 757, 29092, 48884, 74647, 75457, 76267, 92929, 93739, 848848, 1521251, 2985892, 4022204, 4219124, 4251524, 4287824, 5737375, 7875787, 7949497, 27711772, 83155138, 112969211, 123464321

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Keywords

• nonn
• base

References

• J. Meeus, Multibasic palindromes, J. Rec. Math., 18 (No. 3, 1985-1986), 168-173.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Patrick De Geest, Table of n, a(n) for n = 1..65 (terms 1..41 from Robert Israel, terms 42..63 from Robert G. Wilson v)
• M. R. Calandra, Integers which are palindromic in both decimal and binary notation, J. Rec. Math., 18 (No. 1, 1985-1986), 47. (Annotated scanned copy) [With scan of J. Rec. Math. 18.3 (1985), pp. 168-173]
• Patrick De Geest, Palindromic numbers in other bases.

Crossrefs

Cf. A007632, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A099165.

Programs

Maple

ND:= 12;  # to get all terms with <= ND decimal digits
rev10:= proc(n) option remember;
  rev10(floor(n/10)) + (n mod 10)*10^ilog10(n)
end;
for i from 0 to 9 do rev10(i):= i od:
rev3:= proc(n) option remember;
  rev3(floor(n/3)) + (n mod 3)*3^ilog[3](n)
end;
for i from 0 to 2 do rev3(i):= i od:
pali3:= n -> rev3(n) = n;
count:= 1:
A[1]:= 0:
for d from 1 to ND do
  d1:= ceil(d/2);
  for x from 10^(d1-1) to 10^d1 - 1 do
    if d::even then y:= x*10^d1+rev10(x)
    else y:= x*10^(d1-1)+rev10(floor(x/10));
    fi;
    if pali3(y) then
       count:= count+1;
       A[count]:= y;
    fi
  od:
od:
seq(A[i],i=1..count); # Robert Israel, Apr 20 2014

Mathematica

Do[ a = IntegerDigits[n]; b = IntegerDigits[n, 3]; If[a == Reverse[a] && b == Reverse[b], Print[n] ], {n, 0, 10^9} ]
NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]] ]] FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[idn, Ceiling[l/2]], Reverse[ Take[idn, Floor[l/2]] ]]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[idfhn], Drop[ Reverse[ IntegerDigits[idfhn]], Mod[l, 2]] ]]] ]]]; palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; l = {0}; a = 0; Do[a = NextPalindrome[a]; If[ palQ[a, 4], AppendTo[l, a]], {n, 100000}]; l (* Robert G. Wilson v, Sep 30 2004 *)
pal3Q[n_]:=Module[{idn3=IntegerDigits[n,3]},idn3==Reverse[idn3]]; Select[ Range[ 0,1235*10^5],PalindromeQ[#]&&pal3Q[#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 04 2019 *)
Select[Range[0, 10^5],
PalindromeQ[#] && # == IntegerReverse[#, 3] &] (* Robert Price, Nov 09 2019 *)

Python

from itertools import chain
from gmpy2 import digits
A007633_list = sorted([n for n in chain((int(str(x)+str(x)[::-1]) for x in range(1,10**6)),(int(str(x)+str(x)[-2::-1]) for x in range(10**6))) if digits(n,3) == digits(n,3)[::-1]]) # Chai Wah Wu, Nov 23 2014