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A348320 Perfect powers m^k, k >= 2 of palindromes m when m^k is not a palindrome.

%I A348320 #25 Oct 21 2021 01:34:52
%S A348320 16,25,27,32,36,49,64,81,125,128,216,243,256,512,625,729,1024,1089,
%T A348320 1296,1936,2048,2187,2401,3025,3125,4096,4356,5929,6561,7744,7776,
%U A348320 8192,9801,10648,15625,16384,16807,17161,19683,19881,22801,25921,29241,32761,32768,35937,36481,46656
%N A348320 Perfect powers m^k, k >= 2 of palindromes m when m^k is not a palindrome.
%C A348320 Seems to be the "converse" of A348319.
%C A348320 When m is prime, then we get the subsequence A339624.
%C A348320 G. J. Simmons conjectured that there are no palindromes of form n^k for k >= 5 (and n > 1) (see Simmons link p. 98); according to this conjecture, every palindrome^k, k >= 5 is a term.
%H A348320 Gustavus J. Simmons, <a href="/A002778/a002778_2.pdf">Palindromic Powers</a>, J. Rec. Math., Vol. 3, No. 2 (1970), pp. 93-98 [Annotated scanned copy].
%e A348320 216 = 6^3, 1936 = 44^2, 4096 = 8^4, 7776 = 6^5, 35937 = 33^3, 117649 = 7^6 are terms.
%t A348320 seq[max_] := Module[{m = Floor@Sqrt[max], s = {}, n, p}, Do[If[! PalindromeQ[k], Continue[]]; n = Floor@Log[k, max]; Do[If[! PalindromeQ[(p = k^j)], AppendTo[s, p]], {j, 2, n}], {k, 2, m}]; Union[s]]; seq[50000] (* _Amiram Eldar_, Oct 12 2021 *)
%o A348320 (Python)
%o A348320 def ispal(n): s = str(n); return s == s[::-1]
%o A348320 def aupto(limit):
%o A348320     aset, m, mm = set(), 2, 4
%o A348320     while mm <= limit:
%o A348320         if ispal(m):
%o A348320             mk = mm
%o A348320             while mk <= limit:
%o A348320                 if not ispal(mk): aset.add(mk)
%o A348320                 mk *= m
%o A348320         mm += 2*m + 1
%o A348320         m += 1
%o A348320     return sorted(aset)
%o A348320 print(aupto(47000)) # _Michael S. Branicky_, Oct 12 2021
%o A348320 (PARI) ispal(x) = my(d=digits(x)); d == Vecrev(d);
%o A348320 isok(x) = my(q); ispower(x,,&q) && !ispal(x) && ispal(q); \\ _Michel Marcus_, Oct 14 2021
%Y A348320 Cf. A339624, A348319.
%Y A348320 Subsequence of A001597.
%K A348320 nonn,base
%O A348320 1,1
%A A348320 _Bernard Schott_, Oct 12 2021