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a(n) tells which number in square array A083221 (the sieve of Eratosthenes) is at the same position where n is in array A246278. As both arrays have even numbers as their topmost row and primes as their leftmost column, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A083140 is at the same position where n is in the array A246279, as they are the transposes of above two arrays.

The first 7-cycle occurs at: (33 39 63 57 99 81 45) which is mirrored by the cycle (66 78 126 114 198 162 90) with double-size terms.

The cycle which contains 55 as its smallest term, goes as: 55, 65, 95, 185, 425, 325, 205, 455, 395, 1055, 2945, 6035, 30845, ...

while to the other direction (A250246) it goes as: 55, 125, 245, 115, 625, 8575, 40375, ...

The cycle which contains 69 as its smallest term, goes as: 69, 111, 183, 351, 261, 273, 387, 489, 939, 1863, 909, 1161, 981, 1281, 4167, ...

while to the other direction (A250246) it goes as: 69, 135, 87, 105, 225, 207, 231, 195, 525, 1053, 3159, 24909, ...

Permutation A249817 preserves the smallest prime factor of n, i.e., A055396(A249817(n)) = A055396(n), in other words, keeps all the terms that appear on any row of A246278 on the same row of A083221. Permutations in this table are induced by changes that A249817 does onto each row of the latter table, thus permutation on row r of this table can be used to sort row r of A246278 into ascending order. I.e., A246278(r, A(r,c)) = A083221(r,c) [the corresponding row in the Sieve of Eratosthenes, where each row appears in monotone order].

The multi-set of cycle-sizes of permutation A249817 is a disjoint union of cycle-sizes of all permutations in this array. For example, A249817 has a 7-cycle (33 39 63 57 99 81 45) which originates from the 7-cycle (6 7 11 10 17 14 8) of A064216, which occurs as the second row in this table.

On each row, 4 is the first composite number (and the first term less than previous, apart from row 1), and on row n it occurs in position A250474(n). This follows because A001222(A246277(n)) = A001222(n)-1 and because on each row of A083221 (see A083140) all terms between the square of prime (second term on each row) and the first cube (of the same prime, this cube mapping in this array to 4) are nonsquare semiprimes (A006881), this implies that the corresponding terms in this array must be primes.

Also, as the smaller prime factor of the terms on row n of A083221 is constant, A020639(n), and for all i < j: A246277(p_{i} * p_{j}) < A246277(p_i * p_{j+1}), the primes on any row appear in monotone order.

Consider the square array A246278, and also A246275 which is obtained from the former when one is subtracted from each term.

In A246278 the even numbers occur at the top row, and all the rows below that contain only odd numbers, those subsequent terms in each column having been obtained by shifting all primes present in the prime factorization of number immediately above to one larger indices with A003961.

To compute a(n): we do the same process in reverse, by shifting primes in the prime factorization of n+1 step by step to smaller primes, until after k >= 0 such shifts with A064989, the result is even, with the smallest prime present being 2.

We subtract one from this even number and shift the binary expansion of the resulting odd number k positions left (i.e. multiply it with 2^k), which will be the result of a(n).

In the essence, a(n) tells which number in the array A135764 is at the same position where n is in the array A246275. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e., a(2n+1) = 2n+1 for all n.

A055396(n+1) tells on which row of A246275 n is, which is equal to the row of A246278 on which n+1 is.

A246277(n+1) tells in which column of A246275 n is, which is equal to the column of A246278 in which n+1 is.

This is a "doubly-recursed" version of A249817.

For primes p_n, a(p_n) = p_{a(n)}.

The first 7-cycle occurs at: (33 39 63 57 99 81 45), which is mirrored by the cycle (66 78 126 114 198 162 90) with terms double the size and also by the cycle (137 167 307 269 523 419 197), consisting of primes (p_33, p_39, p_63, ...).

Restricted growth sequence transform of function f(n) = A246277(A329044(n)).

For all i, j:

A305800(i) = A305800(j) => a(i) = a(j),

a(i) = a(j) => A329045(i) = A329045(j),

a(i) = a(j) => A329343(i) = A329343(j),

a(i) = a(j) => A329348(i) = A329348(j),

a(i) = a(j) => A329349(i) = A329349(j).

See the comments in A246675. This is otherwise similar permutation, except that after having reached an even number 2m when we have shifted the prime factorization of n+1 k steps towards smaller primes, here, in contrary to A246675, we don't shift the binary representation of the odd number 2m-1, but instead of an odd number (2*a(m))-1 the same number (k) of bit-positions leftward, i.e. multiply it with 2^k.

See also the comments at the inverse permutation A246684.

This is a "more recursed" variant of A249815. Preserves the parity of n.

a(n) tells which number in square array A249741 (the sieve of Eratosthenes minus 1) is at the same position where n is in array A246275. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e. a(2n+1) = 2n+1 for all n. Also, as the leftmost column in both arrays is primes minus one (A006093), they are also among the fixed points.

Equally: a(n) tells which number in array A114881 is at the same position where n is in the array A246273, as they are the transposes of above two arrays.