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A007632 Numbers that are palindromic in bases 2 and 10.
(Formerly M2406)
52
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; history; text; internal format)
OFFSET

1,3

COMMENTS

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

Intersection of A002113 and A006995. - Reinhard Zumkeller, Jan 22 2012, Feb 07 2010

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

Robert G. Wilson v, Charlton Harrison, Ilya Nikulshin, Andrey Astrelin, Table of n, a(n) for n = 1..147  [Terms 97,120,121,127..147 were computed by Ilya Nikulshin & Andrey Astrelin on Dec 30 2015]

Attila Bérczes and Volker Ziegler, On Simultaneous Palindromes, arXiv:1403.0787 [math.NT], 2014 (see p. 9).

P. De Geest, Palindromic numbers beyond base 10

Charlton Harrison, Binary/Decimal Palindromes

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

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 *)

PROG

(Haskell)

a007632 n = a007632_list !! (n-1)

a007632_list = filter ((== 1) . a178225) a002113_list

-- Reinhard Zumkeller, Jan 22 2012

(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

CROSSREFS

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.

Sequence in context: A283003 A141708 A081434 * A117996 A234524 A260681

Adjacent sequences:  A007629 A007630 A007631 * A007633 A007634 A007635

KEYWORD

base,nonn,nice

AUTHOR

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

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

STATUS

approved

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Last modified May 24 17:27 EDT 2017. Contains 286995 sequences.