A002780 Numbers whose cube is a palindrome.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• base
• nonn
• nice

Comments

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.

From Vladimir Shevelev, May 23 2011: (Start)

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)

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), 93-98. [Annotated scanned copy]
• G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Crossrefs

Cf. A002781 (cubes of these numbers).

Programs

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

Extensions

More terms from Patrick De Geest