cp's OEIS Frontend

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.

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, ...).

a(n) tells which number in array A083221 (the sieve of Eratosthenes) is at the same position where n is in Ludic array A255127. As both arrays have A005843 (even numbers) and A016945 as their two topmost rows, 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 A255129, as they are the transposes of above two arrays.

a(n) tells how many columns off A003961(n) is from the column where n is in square array A083221 (Cf. A083140, the sieve of Eratosthenes. The column index of n in that table is given by A078898(n)).

a(n) tells which number in array A255551, constructed from Lucky sieve, is at the same position where n is in array A083221, constructed from the sieve of Eratosthenes. As both arrays have A005843 (even numbers) as their topmost row, this permutation fixes all of them.

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.

The first 7-cycle occurs at: (33 39 63 57 99 81 45), which is mirrored at the cycle (137 167 307 269 523 419 197), consisting of primes (p_33, p_39, p_63, ...).

a(n) tells which number in array A083221, constructed from the sieve of Eratosthenes is at the same position where n is in array A255551 constructed from Lucky sieve. As both arrays have A005843 (even numbers) as their topmost row, this permutation fixes all of them.