A007633 Palindromic in bases 3 and 10. (Formerly M1164)

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

1,3

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A018694 A129661 A018713 * A018777 A130693 A286523

Adjacent sequences: A007630 A007631 A007632 * A007634 A007635 A007636

Keyword

nonn,base

Author

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

Status

approved