A060792 - OEIS

%I #104 Sep 04 2023 10:11:37

%S 0,1,6643,1422773,5415589,90396755477,381920985378904469,

%T 1922624336133018996235,2004595370006815987563563,

%U 8022581057533823761829436662099,392629621582222667733213907054116073,32456836304775204439912231201966254787,428027336071597254024922793107218595973

%N Numbers that are palindromic in bases 2 and 3.

%C a(18) (if it exists) is greater than 3^93. - _Ilya Nikulshin_, Feb 22 2016

%H Ilya Nikulshin, <a href="/A060792/b060792.txt">Table of n, a(n) for n = 1..17</a> (terms a(11)..a(15) from Alan Grimes and _Keith F. Lynch_; a(16) from _Japheth Lim_)

%H Attila Bérczes and Volker Ziegler, <a href="http://arxiv.org/abs/1403.0787">On Simultaneous Palindromes</a>, arXiv:1403.0787 [math.NT], 2014.

%H Alan Grimes, Keith F. Lynch, et al., <a href="http://alt.math.recreational.narkive.com/JnDSktfU/palindrome-numbers">Palindrome numbers</a>.

%H Keith F. Lynch, <a href="http://chesswanks.com/txt/BigDualPalindromes.txt">How I found big dual palindromes</a>.

%e 6643 is a term: since 6643 = 1100111110011_2 = 100010001_3.

%e 1422773 is a term: 1422773 = 101011011010110110101_2 = 2200021200022_3. - _Vladimir Joseph Stephan Orlovsky_, Sep 19 2009

%t 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 *)

%t 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 *)

%o (PARI) ispal(n,b)=my(d=digits(n,b)); d==Vecrev(d)

%o is(n)=ispal(n,2)&&ispal(n,3) \\ _Charles R Greathouse IV_, Jun 17 2014

%o (Python)

%o from itertools import chain

%o from gmpy2 import digits, mpz

%o 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

%o (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

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

%Y Intersection of A006995 and A014190.

%K nonn,base,hard,nice

%O 1,3

%A Ulrich Schimke (ulrschimke(AT)aol.com)

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

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

%E a(9) from Alan Grimes, Dec 16 2013

%E a(10) from _Keith F. Lynch_, Jan 07 2014

%E Term 0 prepended by _Robert G. Wilson v_, Oct 08 2014

%E a(11)-a(15) (from Alan Grimes and _Keith F. Lynch_) added by _Japheth Lim_, Jan 30 2014