OFFSET
2,1
COMMENTS
This is permutation of natural numbers larger than 1.
From Antti Karttunen, Dec 19 2014: (Start)
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252460 gives an inverse permutation. See also A249741.
For navigating in this array:
A055396(n) gives the row number of row where n occurs, and A078898(n) gives its column number, both starting their indexing from 1.
A250469(n) gives the number immediately below n, and when n is an odd number >= 3, A250470(n) gives the number immediately above n. If n is a composite, A249744(n) gives the number immediately left of n.
First cube of each row, which is {the initial prime of the row}^3 and also the first number neither a prime or semiprime, occurs on row n at position A250474(n).
(End)
The n-th row contains the numbers whose least prime factor is the n-th prime: A020639(T(n,k)) = A000040(n). - Franklin T. Adams-Watters, Aug 07 2015
LINKS
EXAMPLE
The top left corner of the array:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26
3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75
5, 25, 35, 55, 65, 85, 95, 115, 125, 145, 155, 175, 185
7, 49, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329
11, 121, 143, 187, 209, 253, 319, 341, 407, 451, 473, 517, 583
13, 169, 221, 247, 299, 377, 403, 481, 533, 559, 611, 689, 767
17, 289, 323, 391, 493, 527, 629, 697, 731, 799, 901, 1003, 1037
19, 361, 437, 551, 589, 703, 779, 817, 893, 1007, 1121, 1159, 1273
23, 529, 667, 713, 851, 943, 989, 1081, 1219, 1357, 1403, 1541, 1633
29, 841, 899, 1073, 1189, 1247, 1363, 1537, 1711, 1769, 1943, 2059, 2117
...
MATHEMATICA
lim = 11; a = Table[Take[Prime[n] Select[Range[lim^2], GCD[# Prime@ n, Product[Prime@ i, {i, 1, n - 1}]] == 1 &], lim], {n, lim}]; Flatten[Table[a[[i, n - i + 1]], {n, lim}, {i, n}]] (* Michael De Vlieger, Jan 04 2016, after Yasutoshi Kohmoto at A083140 *)
PROG
(Scheme, with Antti Karttunen's IntSeq-library)
(define (A083221 n) (if (<= n 1) n (A083221bi (A002260 (- n 1)) (A004736 (- n 1))))) ;; Gives 1 for 1 and then the terms of this square array: (A083221 2) = 2, (A083221 3) = 4, etc.
(define (A083221bi row col) ((rowfun_n_for_A083221 row) col))
(definec (rowfun_n_for_A083221 n) (if (= 1 n) (lambda (n) (+ n n)) (let ((rowfun_of_Esieve (rowfun_n_for_Esieve n)) (prime (A000040 n))) (COMPOSE rowfun_of_Esieve (MATCHING-POS 1 1 (lambda (i) (zero? (modulo (rowfun_of_Esieve i) prime))))))))
(definec (A000040 n) ((rowfun_n_for_Esieve n) 1))
(definec (rowfun_n_for_Esieve n) (if (= 1 n) (lambda (n) (+ 1 n)) (let* ((prevrowfun (rowfun_n_for_Esieve (- n 1))) (prevprime (prevrowfun 1))) (COMPOSE prevrowfun (NONZERO-POS 1 1 (lambda (i) (modulo (prevrowfun i) prevprime)))))))
;; Antti Karttunen, Dec 19 2014
CROSSREFS
Transpose of A083140.
One more than A249741.
Inverse permutation: A252460.
Main diagonal: A083141.
First semiprime in each column occurs at A251717; A251718 & A251719 with additional criteria. A251724 gives the corresponding semiprimes for the latter. See also A251728.
Permutations based on mapping numbers between this array and A246278: A249817, A249818, A250244, A250245, A250247, A250249. See also: A249811, A249814, A249815.
AUTHOR
Yasutoshi Kohmoto, Jun 05 2003
EXTENSIONS
More terms from Hugo Pfoertner, Jun 13 2003
STATUS
approved