A007632 Numbers that are palindromic in bases 2 and 10.
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
Offset: 1
References
- 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).
Links
- Eshed Shaham, Table of n, a(n) for n = 1..175 (Terms 1..147 variously by Robert G. Wilson v, Charlton Harrison, Ilya Nikulshin, Andrey Astrelin)
- Attila Bérczes and Volker Ziegler, On Simultaneous Palindromes, arXiv:1403.0787 [math.NT], 2014 (see p. 9).
- 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 beyond base 10
- Charlton Harrison, Binary/Decimal Palindromes
- Project Euler, Problem 36: Double-base palindromes
- Eshed Shaham, Finding Binary & Decimal Palindromes
Crossrefs
Programs
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Haskell
a007632 n = a007632_list !! (n-1) a007632_list = filter ((== 1) . a178225) a002113_list -- Reinhard Zumkeller, Jan 22 2012
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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
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Maple
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
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 *) -
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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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
Extensions
One more term from George Russell (ger(AT)tzi.de), Nov 20 2000
Two further terms from Harvey P. Dale, Mar 09 2001
Further terms from George Russell (ger(AT)tzi.de), Nov 02 2001
Comments