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A007632
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Numbers that are palindromic in bases 2 and 10.
(Formerly M2406)
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55
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0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717, 7447, 9009, 15351, 32223, 39993, 53235, 53835, 73737, 585585, 1758571, 1934391, 1979791, 3129213, 5071705, 5259525, 5841485, 13500531, 719848917, 910373019, 939474939, 1290880921, 7451111547
(list;
graph;
refs;
listen;
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internal format)
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OFFSET
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1,3
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COMMENTS
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Charlton Harrison found a new record binary-decimal palindrome: 11000101111000010101010110100001110100000100000101110000101101010101000011110100011_2 = 7475703079870789703075747_10 on Dec 01 2001. The binary string contains 83 digits! Since then he has added twenty more terms. - Robert G. Wilson v, Jul 03 2006
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REFERENCES
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M. R. Calandra, Integers which are palindromic in both decimal and binary notation, J. Rec. Math., 18 (No. 1, 1985-1986), 47.
S. Pilpel, Some More Double Palindromic Integers, J. Rec. Math., 18 (1985), 174-176.
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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MAPLE
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N:= 12: # to get all terms <= 10^N
ispal2:= proc(n) local L; if n::even then return false fi;
L:= convert(n, base, 2); evalb(L=ListTools:-Reverse(L)) end proc:
rev10:= proc(n) local L; L:= convert(n, base, 10); add(10^i*L[-i-1], i=0..nops(L)-1) end proc:
pals10:= proc(d) local x, y;
if d::even then [seq(x*10^(d/2)+rev10(x), x=10^(d/2-1)..10^(d/2)-1)]
else [seq(seq(x*10^((d+1)/2)+y*10^((d-1)/2)+rev10(x), y=0..9), x=10^((d-1)/2-1)..10^((d-1)/2)-1)]
fi
end proc:
0, 1, 3, 5, 7, 9, seq(op(select(ispal2, pals10(d))), d=2..N); # Robert Israel, Dec 31 2015
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MATHEMATICA
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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, 2], AppendTo[l, a]], {n, 1000000}]; l (* Robert G. Wilson v, Sep 30 2004 *)
b1=2; b2=10; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 2 10^7}]; lst (* Vincenzo Librandi, Dec 31 2015 *)
Select[Range[0, 10^5], PalindromeQ[#] && # == IntegerReverse[#, 2] &] (* Robert Price, Nov 09 2019 *)
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PROG
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(Haskell)
a007632 n = a007632_list !! (n-1)
a007632_list = filter ((== 1) . a178225) a002113_list
(Python)
from itertools import chain
A007632_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 bin(n)[2:] == bin(n)[:1:-1]]) # Chai Wah Wu, Nov 23 2014
(Magma) [n: n in [0..2*10^7] | Intseq(n, 10) eq Reverse(Intseq(n, 10))and Intseq(n, 2) eq Reverse(Intseq(n, 2))]; // Vincenzo Librandi, Dec 31 2015
(PARI) isok(n) = my(d = digits(n), b=binary(n)); (d == Vecrev(d)) && (b == Vecrev(b)); \\ Michel Marcus, Dec 31 2015
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CROSSREFS
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For number of terms less than or equal to 10^n, see A120764.
Cf. A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A099165.
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KEYWORD
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base,nonn,nice
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AUTHOR
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EXTENSIONS
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One more term from George Russell (ger(AT)tzi.de), Nov 20 2000
Further terms from George Russell (ger(AT)tzi.de), Nov 02 2001
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STATUS
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approved
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