A249821 Square array of permutations: A(row,col) = A246277(A083221(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ... .

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 5, 3, 2, 1, 6, 4, 5, 3, 2, 1, 7, 7, 7, 5, 3, 2, 1, 8, 11, 11, 7, 5, 3, 2, 1, 9, 6, 13, 11, 7, 5, 3, 2, 1, 10, 13, 17, 13, 11, 7, 5, 3, 2, 1, 11, 17, 4, 17, 13, 11, 7, 5, 3, 2, 1, 12, 10, 19, 19, 17, 13, 11, 7, 5, 3, 2, 1, 13, 19, 23, 23, 19, 17, 13, 11, 7, 5, 3, 2, 1, 14, 9, 6, 29, 23, 19, 17, 13, 11, 7, 5, 3, 2, 1, 15, 8, 29, 31, 29, 23, 19, 17, 13, 11, 7, 5, 3, 2, 1

Offset: 1

Author

Antti Karttunen, Nov 06 2014

Keywords

• nonn
• tabl

Comments

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.

Examples

			The top left corner of the array:
1, 2, 3, 4, 5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ...
1, 2, 3, 5, 4,  7, 11,  6, 13, 17, 10, 19,  9,  8, 23, 29, 14, 15, 31, ...
1, 2, 3, 5, 7, 11, 13, 17,  4, 19, 23,  6, 29, 31, 37, 41,  9, 43, 10, ...
1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,  4, 41, 43, 47, 53, 59, ...
...
		

Crossrefs

Inverse permutations can be found from table A249822.

Row k+1 is a left-to-right composition of the first k rows of A251721.

Row 1: A000027 (an identity permutation), Row 2: A064216, Row 3: A249823, Row 4: A249825.

The initial growing part of each row converges towards A008578.

Cf. A001222, A002260, A004736, A006881, A008578, A020639, A083221, A246277, A249817, A250474.

Programs

Scheme

(define (A249821 n) (A249821bi (A002260 n) (A004736 n)))
(define (A249821bi row col) (A246277 (A083221bi row col))) ;; Code for A083221bi given in A083221.

Formula

A(row,col) = A246277(A083221(row,col)).

A001222(A(row,col)) = A001222(A083221(row,col)) - 1. [This follows directly from the properties of A246277.]