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A002780
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Numbers whose cube is a palindrome.
(Formerly M1736 N0688)
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11
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0, 1, 2, 7, 11, 101, 111, 1001, 2201, 10001, 10101, 11011, 100001, 101101, 110011, 1000001, 1001001, 1100011, 10000001, 10011001, 10100101, 11000011, 100000001, 100010001, 100101001, 101000101, 110000011, 1000000001, 1000110001
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OFFSET
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1,3
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COMMENTS
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a(8)=2201 is the only known non-palindromic rootnumber.
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
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.
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).
Does there exist a term a(n)>10^17+10^16+11 which is in none of these classes?
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).
One can prove that the sequence contains no term (other than 2201) with sum of digits = 5. (End)
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]
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PROG
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(PARI) isok(k) = my(d=digits(k^3)); Vecrev(d) == d; \\ Michel Marcus, Aug 02 2022
(Python)
def ispal(s): return s == s[::-1]
def ok(n): return ispal(str(n**3))
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CROSSREFS
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Cf. A002781 (cubes of these numbers).
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KEYWORD
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base,nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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