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 A056810 Numbers whose fourth power is a palindrome. 3
 0, 1, 11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 100000000001 (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 last two 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 *) PROG (Python) def ispal(n): s = str(n); return s == s[::-1] def afind(limit):     for k in range(limit+1):         if ispal(k**4): print(k, end=", ") afind(10000001) # Michael S. Branicky, Sep 05 2021 CROSSREFS Cf. A186080. Sequence in context: A031997 A116098 A116129 * A000533 A147759 A147757 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 a(12)-a(13) from Michael S. Branicky, Sep 05 2021 STATUS approved

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Last modified September 27 19:37 EDT 2022. Contains 357063 sequences. (Running on oeis4.)