This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A056810 #56 Jun 13 2025 12:13:04
%S A056810 0,1,11,101,1001,10001,100001,1000001,10000001,100000001,1000000001,
%T A056810 10000000001,100000000001
%N A056810 Numbers whose fourth power is a palindrome.
%C A056810 Suppose a number is of the form a=10...01 (see A000533) 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
%C A056810 The sequence contains no term with digit sum 3. - _Vladimir Shevelev_, May 23 2011. Proof: There are four possibilities for n:
%C A056810 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.
%C A056810 In the last two cases n^4 is trivially not a palindrome.
%C A056810 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)
%C A056810 which cannot be a palindrome.
%C A056810 If r=1, we have 1+8*10+...9*10^4+10^5 which also is not a palindrome.
%C A056810 The proof for the first case is similar. QED - _Vladimir Shevelev_, Oct 24 2015
%C A056810 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
%H A056810 G. J. Simmons, <a href="/A002778/a002778_2.pdf">Palindromic powers</a>, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]
%H A056810 G. J. Simmons, <a href="/A002778/a002778.pdf">On palindromic squares of non-palindromic numbers</a>, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]
%t A056810 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 *)
%o A056810 (Python)
%o A056810 def ispal(n): s = str(n); return s == s[::-1]
%o A056810 def afind(limit):
%o A056810 for k in range(limit+1):
%o A056810 if ispal(k**4): print(k, end=", ")
%o A056810 afind(10000001) # _Michael S. Branicky_, Sep 05 2021
%Y A056810 Cf. A186080.
%K A056810 nonn,base,more
%O A056810 1,3
%A A056810 _Robert G. Wilson v_, Aug 21 2000
%E A056810 a(11) from _Robert G. Wilson v_, Oct 23 2015
%E A056810 a(12)-a(13) from _Michael S. Branicky_, Sep 05 2021