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A075786 Palindromic perfect powers.

%I A075786 #32 Aug 27 2021 02:16:51
%S A075786 1,4,8,9,121,343,484,676,1331,10201,12321,14641,40804,44944,69696,
%T A075786 94249,698896,1002001,1030301,1234321,1367631,4008004,5221225,6948496,
%U A075786 100020001,102030201,104060401,121242121,123454321,125686521,400080004,404090404,522808225,617323716,942060249
%N A075786 Palindromic perfect powers.
%C A075786 Up to 10^12, there are only 43 perfect powers which are palindromic.
%C A075786 The sequence is infinite, for instance it contains (10^k+1)^2. - _Emmanuel Vantieghem_, Sep 29 2017
%C A075786 Conjecture: there are no palindromic perfect powers with prime exponent > 3. - _Chai Wah Wu_, Aug 26 2021
%H A075786 Chai Wah Wu, <a href="/A075786/b075786.txt">Table of n, a(n) for n = 1..2001</a> (n = 1..176 from Michael S. Branicky, n = 177..504 from David A. Corneth)
%e A075786 343 = 7^3 is a term as it is a palindrome and a perfect power. - _David A. Corneth_, Mar 23 2021
%t A075786 a = {}; Do[q = IntegerDigits[n]; p = FromDigits[ Join[ q, Reverse[ Drop[q, -1]]]]; If[ Apply[ GCD, Last[ Transpose[ FactorInteger[p]]]] > 1, a = Append[a, p]]; p = FromDigits[ Join[ q, Reverse[q]]]; If[ Apply[ GCD, Last[ Transpose[ FactorInteger[p]]]] > 1, a = Append[a, p]], {n, 1, 10^5}]
%o A075786 (Python)
%o A075786 from math import isqrt
%o A075786 def ispal(n): s = str(n); return s == s[::-1]
%o A075786 def athrough(digits):
%o A075786   found, limit = {1}, 10**digits
%o A075786   for k in range(2, isqrt(limit) + 1):
%o A075786     kpow = k*k
%o A075786     while kpow < limit:
%o A075786       if ispal(kpow): found.add(kpow)
%o A075786       kpow *= k
%o A075786   return sorted(found)
%o A075786 print(athrough(9)) # _Michael S. Branicky_, Mar 23 2021
%Y A075786 Cf. A001597, A002113, A076443.
%K A075786 nonn,base
%O A075786 1,2
%A A075786 _Zak Seidov_, Oct 10 2002
%E A075786 Edited and extended by _Robert G. Wilson v_, Oct 11 2002
%E A075786 More terms from _David A. Corneth_, Mar 24 2021
%E A075786 b-file corrected and extended by _Chai Wah Wu_, Aug 26 2021