A060792 Numbers that are palindromic in bases 2 and 3.

0, 1, 6643, 1422773, 5415589, 90396755477, 381920985378904469, 1922624336133018996235, 2004595370006815987563563, 8022581057533823761829436662099, 392629621582222667733213907054116073, 32456836304775204439912231201966254787, 428027336071597254024922793107218595973

Offset: 1

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

• nonn
• base
• hard
• nice

Comments

a(18) (if it exists) is greater than 3^93. - Ilya Nikulshin, Feb 22 2016

Examples

			6643 is a term: since 6643 = 1100111110011_2 = 100010001_3.
1422773 is a term: 1422773 = 101011011010110110101_2 = 2200021200022_3. - _Vladimir Joseph Stephan Orlovsky_, Sep 19 2009
		

Links

• Ilya Nikulshin, Table of n, a(n) for n = 1..17 (terms a(11)..a(15) from Alan Grimes and _Keith F. Lynch_; a(16) from _Japheth Lim_)
• Attila Bérczes and Volker Ziegler, On Simultaneous Palindromes, arXiv:1403.0787 [math.NT], 2014.
• Alan Grimes, Keith F. Lynch, et al., Palindrome numbers.
• Keith F. Lynch, How I found big dual palindromes.

Crossrefs

a(3) = A048268(2) = A056749(3).

Intersection of A006995 and A014190.

Programs

Magma

[n: n in [0..2*10^7] | Intseq(n, 3) eq Reverse(Intseq(n, 3))and Intseq(n, 2) eq Reverse(Intseq(n, 2))]; // Vincenzo Librandi, Feb 24 2016

Mathematica

pal2Q[n_Integer] := IntegerDigits[n, 2] == Reverse[IntegerDigits[n, 2]]; pal3Q[n_Integer] := IntegerDigits[n, 3] == Reverse[IntegerDigits[n, 3]]; A060792 = {}; Do[If[pal2Q[n] && pal3Q[n], AppendTo[A060792, n]], {n, 12!}]; A060792 (* Vladimir Joseph Stephan Orlovsky, Sep 19 2009 *)
b1=2; b2=3; 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, Feb 24 2016 *)

PARI

ispal(n,b)=my(d=digits(n,b)); d==Vecrev(d)
is(n)=ispal(n,2)&&ispal(n,3) \\ Charles R Greathouse IV, Jun 17 2014

Python

from itertools import chain
from gmpy2 import digits, mpz
A060792 = [int(n,2) for n in chain(map(lambda x:bin(x)[2:]+bin(x)[2:][::-1],range(1,2**16)),map(lambda x:bin(x)[2:]+bin(x)[2:][-2::-1], range(1,2**16))) if mpz(int(n,2)).digits(3) == mpz(int(n,2)).digits(3)[::-1]] # Chai Wah Wu, Aug 12 2014

Extensions

a(7) found by François Boisson, using a Caml program running on an AMD-64 machine. - Bruno Petazzoni, program co-author, Jan 31 2006

a(8) from the same source, May 26 2006

a(9) from Alan Grimes, Dec 16 2013

a(10) from Keith F. Lynch, Jan 07 2014

Term 0 prepended by Robert G. Wilson v, Oct 08 2014

a(11)-a(15) (from Alan Grimes and Keith F. Lynch) added by Japheth Lim, Jan 30 2014