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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. 5
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, 47, 49, 53, 59, 61, 64, 65, 67, 70, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 133, 137, 139, 149, 151, 154, 157, 163, 165, 167, 169 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

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.

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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

EXAMPLE

1 is present because it has an empty factorization, so both sequences are empty, thus palindromes.

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.

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.

8 = p_1^3 is present, as both the deltas of prime indices (1) and the exponents (3) form a palindrome.

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.

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.

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.

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.

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.

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.

PROG

(Scheme, with Antti Karttunen's IntSeq-library)

(define A242417 (FIXED-POINTS 1 1 A242419))

CROSSREFS

Fixed points of A242419. Intersection of A242413 and A242414.

Subsequences: A000961, A002110.

Cf. A243058, A243068, A242421, A088902, A241912.

Sequence in context: A085869 A068892 A106801 * A225657 A175772 A124868

Adjacent sequences:  A242414 A242415 A242416 * A242418 A242419 A242420

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 20 2014

STATUS

approved

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Last modified February 22 17:35 EST 2019. Contains 320400 sequences. (Running on oeis4.)