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A056810 Numbers n such that the fourth power of n is a palindrome. 2
0, 1, 11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Suppose a number is of the form a=10...01 then a^2=10..020..01, so a^2 is always a palindrome. a^3=10..030..030..01, so a^3 is always a palindrome. Similarly we also have a^4=10..040..060..040..01, so a^4 is always a palindrome. However, a^5 is in general not a palindrome, for example 101^5=10510100501. - Dmitry Kamenetsky, Apr 17 2009

The sequence contains no term with digit sum 3. - Vladimir Shevelev, May 23 2011.  Proof: There are four possibilities for n:

1) 1+10^k+10^m, 0<k<m, 2) 1+2*10^r, r>0, 3) 2+10^s, s>0, 4) 3*10^t, t>=0.

In the two last cases n^4 is trivially not a palindrome.

For r>=2, in the second case we have n^4 = (1 + 2*10^r)^4 = 1 + 8*10^r + 4*10^(2*r) + 2*10^(2*r + 1) + 2*10^(3*r) + 3*10^(3*r + 1) + 6*10^(4*r) + 10^(4*r + 1)

which cannot be a palindrome.

If r=1, we have 1+8*10+...9*10^4+10^5 which also is not a palindrome.

The proof for the first case is similar. QED  Vladimir Shevelev, Oct 24 2015

Does every term have the structure 100...0001? Referring to the Simmons (1972) paper, we can also ask, if n is a number whose cube is a palindrome in base 4, must the base-4 expansion of n have the form 100...0001? - N. J. A. Sloane, Oct 22 2015

LINKS

Table of n, a(n) for n=1..11.

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]

MATHEMATICA

palQ[n_] := Block[{}, Reverse[idn = IntegerDigits@ n] == idn]; k = 0; lst = {}; While[k < 1000000002, If[ palQ[k^4], AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Oct 23 2015 *)

CROSSREFS

Cf. A186080.

Sequence in context: A191420 A118937 A031997 * A116098 A116129 A000533

Adjacent sequences:  A056807 A056808 A056809 * A056811 A056812 A056813

KEYWORD

nonn,base,more

AUTHOR

Robert G. Wilson v, Aug 21 2000

EXTENSIONS

a(11) from Robert G. Wilson v, Oct 23 2015

STATUS

approved

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Last modified June 28 23:50 EDT 2016. Contains 274272 sequences.