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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, 00, 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 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 July 6 17:36 EDT 2020. Contains 335479 sequences. (Running on oeis4.)