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A242414 Numbers whose prime factorization viewed as a tuple of nonzero powers is palindromic.

%I A242414 #19 Feb 09 2025 08:10:48
%S A242414 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,30,31,
%T A242414 32,33,34,35,36,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,64,
%U A242414 65,66,67,69,70,71,73,74,77,78,79,81,82,83,85,86,87,89,90,91,93,94,95,97,100
%N A242414 Numbers whose prime factorization viewed as a tuple of nonzero powers is palindromic.
%C A242414 The fixed points of permutation A069799.
%C A242414 Differs from its subsequence, A072774, Powers of squarefree numbers, for the first time at n=68, as here a(68) = 90 is included, as 90 = p_1^1 * p_2^2 * p_3^1 has a palindromic tuple of exponents, even although not all of them are identical.
%C A242414 Differs from its another subsequence, A236510, in that, although numbers like 42 = 2^1 * 3^1 * 5^0 * 7^1, with a non-palindromic exponent-tuple (1,1,0,1) are excluded from A236510, it is included in this sequence, because here only the nonzero exponents are considered, and (1,1,1) is a palindrome.
%C A242414 Differs from A085924 in that as that sequence is subtly base-dependent, it excludes 1024 (= 2^10), as then the only exponent present, 10, and thus also its concatenation, "10", is not a palindrome when viewed in decimal base. On the contrary, here a(691) = 1024.
%H A242414 Antti Karttunen, <a href="/A242414/b242414.txt">Table of n, a(n) for n = 1..10000</a>
%e A242414 As 1 has an empty factorization, (), which also is a palindrome, 1 is present.
%e A242414 As 42 = 2 * 3 * 7 = p_1^1 * p_2^1 * p_4^1, and (1,1,1) is palindrome, 42 is present.
%e A242414 As 90 = 2 * 9 * 5 = p_1^1 * p_2^2 * p_3^1, and (1,2,1) is palindrome, 90 is present.
%e A242414 Any prime power (A000961) is present, as such numbers have a factorization p^e (e >= 1), and any singleton sequence (e) by itself forms a palindrome.
%t A242414 Select[Range[100], PalindromeQ[FactorInteger[#][[All, 2]]]&] (* _Jean-François Alcover_, Feb 09 2025 *)
%o A242414 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A242414 (define A242414 (FIXED-POINTS 1 1 A069799))
%Y A242414 Fixed points of A069799.
%Y A242414 Complement: A242416.
%Y A242414 A000961, A072774 and A236510 are subsequences.
%Y A242414 Cf. A242418, A085924.
%K A242414 nonn
%O A242414 1,2
%A A242414 _Antti Karttunen_, May 30 2014