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A348429 Perfect powers m^k, m >= 1, k >= 2 such that m and m^k both are palindromes.

%I A348429 #41 Oct 21 2021 10:40:58
%S A348429 1,4,8,9,121,343,484,1331,10201,12321,14641,40804,44944,1002001,
%T A348429 1030301,1234321,1367631,4008004,100020001,102030201,104060401,
%U A348429 121242121,123454321,125686521,400080004,404090404,1003003001,10000200001,10221412201,12102420121,12345654321,40000800004
%N A348429 Perfect powers m^k, m >= 1, k >= 2 such that m and m^k both are palindromes.
%C A348429 Complement of A348319 relative to the positive perfect powers A001597.
%C A348429 This sequence is infinite since each square (10^m+1)^2 is a term for m >= 0 and A033934 is a subsequence.
%C A348429 Observation: terms always contain an odd number of digits.
%C A348429 For k = 2, subsequence of palindromes whose square root is a palindrome is A057136 (see A057135).
%C A348429 For k = 3, except for 2201^3 = 10662526601, all known palindromic cubes have a palindromic rootnumber (see A002780 and A002781).
%C A348429 For k = 4, all known integers whose fourth power is a palindrome are also palindromes (see A056810 and subsequence A186080).
%C A348429 For k >= 5, G. J. Simmons conjectured there are no palindromes of the form m^k for k >= 5 and m > 1 (see Simmons link p. 98); according to this conjecture, all the terms are of the form (palindrome)^k, with 2 <= k <= 4.
%H A348429 Michael S. Branicky, <a href="/A348429/b348429.txt">Table of n, a(n) for n = 1..1024</a> (all terms with <= 40 digits)
%H A348429 Michael S. Branicky, <a href="/A348429/a348429.txt">Python program</a>
%H A348429 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 A348429 First few terms are equal to 1, 2^2, 2^3, 3^2, 11^2, 7^3, 22^2, 11^3, 101^2, 111^2, 11^4 = 121^2, 202^2, 212^2, 1001^2, 101^3, 1111^2, 111^3.
%t A348429 Block[{n = 10^6, nn, s}, s = Select[Range[2, n], PalindromeQ]; nn = Max[s]^2; {1}~Join~Union@ Reap[Table[Do[If[PalindromeQ[m^k], Sow[m^k]], {k, 2, Log[m, nn]}], {m, s}]][[-1, -1]]] (* _Michael De Vlieger_, Oct 18 2021 *)
%o A348429 (Python) # see link for faster version
%o A348429 def ispal(n): s = str(n); return s == s[::-1]
%o A348429 def aupto(limit):
%o A348429     aset, m, mm = {1}, 2, 4
%o A348429     while mm <= limit:
%o A348429         if ispal(m):
%o A348429             mk = mm
%o A348429             while mk <= limit:
%o A348429                 if ispal(mk): aset.add(mk)
%o A348429                 mk *= m
%o A348429         mm += 2*m + 1
%o A348429         m += 1
%o A348429     return sorted(aset)
%o A348429 print(aupto(10**11)) # _Michael S. Branicky_, Oct 18 2021
%o A348429 (PARI) ispal(x) = my(d=digits(x)); d == Vecrev(d); \\ A002113
%o A348429 isok(m) = if (m==1, return (1)); my(p); ispal(m) && ispower(m, , &p) && ispal(p); \\ _Michel Marcus_, Oct 19 2021
%o A348429 (PARI) ispal(x) = my(d=digits(x)); d == Vecrev(d); \\ A002113
%o A348429 lista(nn) = {my(list = List(1)); for (k=2, sqrtint(nn), if (ispal(k), my(q = k^2); until (q > nn, if (ispal(q), listput(list, q)); q *= k;););); vecsort(list,,8);} \\ _Michel Marcus_, Oct 20 2021
%Y A348429 Cf. A001597, A002113, A033934, A056810, A186080, A348319, A348320.
%K A348429 nonn,base
%O A348429 1,2
%A A348429 _Bernard Schott_, Oct 18 2021