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A242417 Numbers in whose prime factorization both the first differences of indices of distinct primes and their exponents form a palindrome; fixed points of A242419.

%I A242417 #33 May 06 2018 00:47:35
%S A242417 1,2,3,4,5,6,7,8,9,11,13,16,17,19,21,23,25,27,29,30,31,32,36,37,41,43,
%T A242417 47,49,53,59,61,64,65,67,70,71,73,79,81,83,89,90,97,101,103,107,109,
%U A242417 113,121,125,127,128,131,133,137,139,149,151,154,157,163,165,167,169
%N A242417 Numbers in whose prime factorization both the first differences of indices of distinct primes and their exponents form a palindrome; fixed points of A242419.
%C A242417 Numbers that are fixed by the permutation A242419, i.e., for which A242419(n) = n. Also, numbers that are fixed by both A069799 and A242415.
%C A242417 Number n is present if its prime factorization n = p_a^e_a * p_b^e_b * p_c^e_c * ... * p_i^e_i * p_j^e_j * p_k^e_k where a < b < c < ... < i < j < k, satisfies the condition, that both the first differences of prime indices (a-0, b-a, c-b, ..., j-i, k-j) and the respective exponents (e_a, e_b, e_c, ... , e_i, e_j, e_k) form a palindrome.
%C A242417 More formally, numbers n whose prime factorization is either of the form p^e (p prime, e >= 0), i.e., one of the terms of A000961, or of the form p_i1^e_i1 * p_i2^e_i2 * p_i3^e_i3 * ... * p_i_{k-1}^e_{i_{k-1}} * p_{i_k}^e_{i_k}, where p_i1 < p_i2 < ... < p_i_{k-1} < p_k are distinct primes (sorted into ascending order) in the prime factorization of n, and e_i1 .. e_{i_k} are their nonzero exponents (here k =  A001221(n) and i_k = A061395(n), the index of the largest prime present), and the indices of primes satisfy the relation that for each index i_1 <= i_j < i_k present, the index i_{k-j} is also present, and the exponents e_{i_j} and e_{i_{(k-j)+1}} are equal.
%H A242417 Antti Karttunen, <a href="/A242417/b242417.txt">Table of n, a(n) for n = 1..10000</a>
%e A242417 1 is present because it has an empty factorization, so both sequences are empty, thus palindromes.
%e A242417 3 = p_2^1 is present, as both the sequence of the first differences (deltas) of prime indices (2-0) = (2) and the exponents (1) are palindromes.
%e A242417 6 = p_1^1 * p_2^1 is present, as both the deltas of prime indices (1-0, 2-1) = (1,1) and the exponents (1,1) form a palindrome.
%e A242417 8 = p_1^3 is present, as both the deltas of prime indices (1) and the exponents (3) form a palindrome.
%e A242417 300 = 4*3*25 = p_1^2 * p_2^1 * p_3^2 is present, as both the deltas of prime indices (1-0, 2-1, 3-2) = (1,1,1) 1, 2 and the exponents (2,1,2), form a palindrome.
%e A242417 144 = 2^4 * 3^2 = p_1^4 * p_2^2 is NOT present, as although the deltas of prime indices (1-0, 2-1) = (1,1) are palindrome, the sequence of exponents (4,2) do NOT form a palindrome.
%e A242417 441 = 9*49 = p_2^2 * p_4^2 is present, as both the deltas of prime indices (2-0, 4-2) = (2,2) and the exponents (2,2) form a palindrome.
%e A242417 30030 = 2*3*5*7*11*13 = p_1 * p_2 * p_3 * p_4 * p_5 * p_6 is present, as the exponents are all ones, and the deltas of indices, (6-5,5-4,4-3,3-2,2-1,1-0) = (1,1,1,1,1,1) likewise are all ones, thus both sequences form a palindrome. This is true for all primorial numbers, A002110.
%e A242417 47775 = 3*5*5*7*7*13 = p_2^1 * p_3^2 * p_4^2 * p_6^1 is present, as the deltas of indices (6-4,4-3,3-2,2-0) = (2,1,1,2) and the exponents (1,2,2,1) both form a palindrome.
%e A242417 90000 = 2*2*2*2*3*3*5*5*5*5 = p_1^4 * p_2^2 * p_3^4 is present, as the deltas of indices (3-2,2-1,1-0) = (1,1,1) and the exponents (4,2,4) both form a palindrome.
%o A242417 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A242417 (define A242417 (FIXED-POINTS 1 1 A242419))
%Y A242417 Fixed points of A242419. Intersection of A242413 and A242414.
%Y A242417 Subsequences: A000961, A002110.
%Y A242417 Cf. A243058, A243068, A242421, A088902, A241912.
%K A242417 nonn
%O A242417 1,2
%A A242417 _Antti Karttunen_, May 20 2014