

A002780


Numbers whose cube is a palindrome.
(Formerly M1736 N0688)


11



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

1,3


COMMENTS

a(8)=2201 is the only known nonpalindromic rootnumber.
There are no further nonpalindromic 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 nonpalindromic terms in the range 10^15 to 10^20 with digits from the set {0,1,2}.  Hans Havermann, May 18 2011.
From Vladimir Shevelev, May 23 2011: (Start)
Using the table by NoeDe 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)


REFERENCES

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).


LINKS

T. D. Noe, Table of n, a(n) for n = 1..89 (from De Geest)
Patrick De Geest, Palindromic Cubes
G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 9398. [Annotated scanned copy]
G. J. Simmons, On palindromic squares of nonpalindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 1119. [Annotated scanned copy]


PROG

(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))
print([k for k in range(10**7) if ok(k)]) # Michael S. Branicky, Aug 02 2022


CROSSREFS

Cf. A002781 (cubes of these numbers).
Sequence in context: A349709 A135066 A085315 * A069885 A069748 A064441
Adjacent sequences: A002777 A002778 A002779 * A002781 A002782 A002783


KEYWORD

base,nonn,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Patrick De Geest


STATUS

approved



