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A007632 Numbers that are palindromic in bases 2 and 10.
(Formerly M2406)
55
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
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]
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 *)
Select[Range[0, 10^5], PalindromeQ[#] && # == IntegerReverse[#, 2] &] (* Robert Price, Nov 09 2019 *)
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.
Sequence in context: A329590 A141708 A081434 * A117996 A234524 A260681
KEYWORD
base,nonn,nice
AUTHOR
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 March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)