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A242413 Numbers in whose prime factorization the first differences of indices of distinct primes form a palindrome; fixed points of A242415.

%I A242413 #25 May 06 2018 02:24:00
%S A242413 1,2,3,4,5,6,7,8,9,11,12,13,16,17,18,19,21,23,24,25,27,29,30,31,32,36,
%T A242413 37,41,43,47,48,49,53,54,59,60,61,63,64,65,67,70,71,72,73,79,81,83,89,
%U A242413 90,96,97,101,103,107,108,109,113,120,121,125,127,128,131,133,137,139,140,144
%N A242413 Numbers in whose prime factorization the first differences of indices of distinct primes form a palindrome; fixed points of A242415.
%C A242413 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 the first differences of prime indices (a-0, b-a, c-b, ..., j-i, k-j) form a palindrome.
%C A242413 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_j < i_k present, the index i_{k-j} is also present.
%H A242413 Antti Karttunen, <a href="/A242413/b242413.txt">Table of n, a(n) for n = 1..10000</a>
%e A242413 1 is present because it has an empty factorization, so both the sequence of the prime indices and their first differences are empty, and empty sequences are palindromes as well.
%e A242413 12 = 2*2*3 = p_1^2 * p_2 is present, as the first differences (deltas) of prime indices (1-0, 2-1) = (1,1) form a palindrome.
%e A242413 60 = 2*2*3*5 = p_1^2 * p_2 * p_3 is present, as the deltas of prime indices (1-0, 2-1, 3-2) = (1,1,1) form a palindrome.
%e A242413 61 = p_18 is present, as the deltas of prime indices, (18), form a palindrome.
%e A242413 144 = 2^4 * 3^2 = p_1^4 * p_2^2 is present, as the deltas of prime indices (1-0, 2-1) = (1,1) form a palindrome.
%e A242413 Also, any of the cases mentioned in the Example section of A242417 as being present there, are also present in this sequence.
%o A242413 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A242413 (define A242413 (FIXED-POINTS 1 1 A242415))
%Y A242413 Fixed points of A242415.
%Y A242413 Differs from A243068 for the first time at n=36, where a(36)=60, while A243068(36)=61.
%Y A242413 Cf. A000961, A242417 (subsequences), A242414, A243058, A242421, A088902, A241912.
%K A242413 nonn
%O A242413 1,2
%A A242413 _Antti Karttunen_, May 31 2014