A069748 Numbers k such that k and k^3 are both palindromes.

0, 1, 2, 7, 11, 101, 111, 1001, 10001, 10101, 11011, 100001, 101101, 110011, 1000001, 1001001, 1100011, 10000001, 10011001, 10100101, 11000011, 100000001, 100010001, 100101001, 101000101, 110000011, 1000000001, 1000110001, 1010000101, 1100000011, 10000000001

Offset: 1

Author

Joseph L. Pe, Apr 22 2002

Keywords

• base
• nonn

Comments

For an arithmetical function f, call the pairs (x,y) such that y = f(x) and x, y are palindromes the "palinpairs" of f. {a(n)} is then the sequence of abscissae of palinpairs of f(n) = n^3.

Perhaps this sequence is the same as A002780, except for 2201. - Dmitry Kamenetsky, Apr 16 2009

For n >= 5, there are no terms with digit sum 5. Conjecture: all terms belong to one of 3 disjoint classes of the following forms: 10^k+1, 10^(2*t)+10^t+1, t > 0, and (10^u+1)*(10^v+1), u,v > 0, with digit sums 2, 3 and 4 correspondingly. - Vladimir Shevelev, May 31 2011

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..117
• Vladimir Shevelev, Re: numbers whose cube is a palindrome, seqfan list, May 25 2011

Crossrefs

Intersection of A002113 and A002780.

Programs

Mathematica

isPalin[n_] := (n == FromDigits[Reverse[IntegerDigits[n]]]); Do[m = n^3; If[isPalin[n] && isPalin[m], Print[{n, m}]], {n, 1, 10^6}]

PARI

ispal(n) = my(d=digits(n)); d == Vecrev(d);
isok(n) = ispal(n) && ispal(n^3); \\ Michel Marcus, Dec 16 2018

Extensions

a(29) and beyond from Michael S. Branicky, Aug 06 2022