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%I #41 Mar 31 2021 03:53:58
%S 16,25,27,32,49,64,81,125,128,243,256,512,625,729,1024,2048,2187,2401,
%T 3125,4096,6561,8192,15625,16384,16807,17161,19683,22801,32761,32768,
%U 36481,59049,65536,78125,97969,117649,124609,131072,139129,146689,161051,177147,262144
%N Perfect powers p^k, k >= 2 of palindromic primes p when p^k is not a palindrome.
%C Equivalently: numbers m with only one prime factor such that the LCM of their palindromic divisors is neither 1 nor m: subsequence of A334392.
%C G. J. Simmons conjectured there are no palindromes of form n^k for k >= 5 (and n > 1) (see Simmons p. 98). According to this conjecture, these perfect powers are terms: {2^k, k>=4}, {3^k, k>=3}, {5^k, k>=2}, {7^k, k=2 and k>=4}, {11^k, k>=5}, {101^k, k>= 5}, {131^k, k>=2}, ...
%C From a(1) = 16 to a(17) = 2187, the data is the same as A056781(10) until A056781(26), then a(18) = 2401 and A056781(27) = 4096.
%D Murray S. Klamkin, Problems in applied mathematics: selections from SIAM review, (1990), p. 520.
%H Gustavus J. Simmons, <a href="https://oeis.org/A002778/a002778_2.pdf">Palindromic Powers</a>, J. Rec. Math., 3 (No. 2, 1970), 93-98 [Annotated scanned copy].
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Palindromic_number#Perfect_powers">Palindromic number, Perfect Powers</a>.
%e 5^2 = 25, 2^6 = 64, 3^4 = 81 are terms.
%e 7^2 = 49 is a term, 7^3 = 343 is not a term, and 7^4 = 2401 is a term.
%e 101^2 = 10201 and 11^4 = 14641 are not terms.
%t q[n_] := Module[{f = FactorInteger[n]}, Length[f] == 1 && f[[1, 2]] > 1 && PalindromeQ[f[[1, 1]]]]; Select[Range[10^5], !PalindromeQ[#] && q[#] &] (* _Amiram Eldar_, Dec 10 2020 *)
%o (PARI) ispal(n) = my(d=digits(n)); Vecrev(d)==d;
%o isok(k) = my(p); isprimepower(k, &p) && isprime(p) && ispal(p) &&!ispal(k); \\ _Michel Marcus_, Dec 10 2020
%Y Intersection of A025475 and A334392.
%Y Cf. A087990, A087999, A334139, A334391.
%Y Subsequences: A000079 \ {1,2,4,8}, A000244 \ {1,3,9}, A000351 \ {1,5}, A000420 \ {1,7,343}, A001020 \ {1,11,121,1331,14641}, A096884 \ {1,101, 10201, 1030301, 104060401}.
%K nonn,base
%O 1,1
%A _Bernard Schott_, Dec 10 2020
%E More terms from _Amiram Eldar_, Dec 10 2020