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A007633
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Palindromic in bases 3 and 10.
(Formerly M1164)
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41
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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REFERENCES
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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).
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LINKS
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Patrick De Geest, Table of n, a(n) for n = 1..65 (first 41 terms 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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,base
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AUTHOR
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N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v
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STATUS
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approved
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