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%I M1736 N0688 #42 Aug 02 2022 15:39:51
%S 0,1,2,7,11,101,111,1001,2201,10001,10101,11011,100001,101101,110011,
%T 1000001,1001001,1100011,10000001,10011001,10100101,11000011,
%U 100000001,100010001,100101001,101000101,110000011,1000000001,1000110001
%N Numbers whose cube is a palindrome.
%C a(8)=2201 is the only known non-palindromic rootnumber.
%C There are no further non-palindromic terms (other than 2201) up to 10^11. - _Matevz Markovic_, Apr 04 2011. There are none up to 10^15, by direct search. - _Charles R Greathouse IV_, May 16 2011
%C There are no non-palindromic terms in the range 10^15 to 10^20 with digits from the set {0,1,2}. - _Hans Havermann_, May 18 2011.
%C From _Vladimir Shevelev_, May 23 2011: (Start)
%C Using the table by Noe-De Geest, I noticed that all numbers {a(n)=A002780(n); 11<=a(n)<=10^17+10^16+11}, except 2201, allow a partition into 3 disjoint classes of terms of the following forms: 10^k+1, 10^(2*k)+10^k+1, and (10^u+1)*(10^v+1).
%C Does there exist a term a(n)>10^17+10^16+11 which is in none of these classes?
%C If there is no such term, then we conclude that the sum of digits of a(n) does not exceed 4 (more exactly, it is i+1 where i is the number of class).
%C One can prove that the sequence contains no term (other than 2201) with sum of digits = 5. (End)
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002780/b002780.txt">Table of n, a(n) for n = 1..89</a> (from De Geest)
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/cube.htm">Palindromic Cubes</a>
%H G. J. Simmons, <a href="/A002778/a002778_2.pdf">Palindromic powers</a>, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]
%H G. J. Simmons, <a href="/A002778/a002778.pdf">On palindromic squares of non-palindromic numbers</a>, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]
%o (PARI) isok(k) = my(d=digits(k^3)); Vecrev(d) == d; \\ _Michel Marcus_, Aug 02 2022
%o (Python)
%o def ispal(s): return s == s[::-1]
%o def ok(n): return ispal(str(n**3))
%o print([k for k in range(10**7) if ok(k)]) # _Michael S. Branicky_, Aug 02 2022
%Y Cf. A002781 (cubes of these numbers).
%K base,nonn,nice
%O 1,3
%A _N. J. A. Sloane_
%E More terms from _Patrick De Geest_
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