A083221 Sieve of Eratosthenes arranged as an array and read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 21, 35, 49, 11, 12, 27, 55, 77, 121, 13, 14, 33, 65, 91, 143, 169, 17, 16, 39, 85, 119, 187, 221, 289, 19, 18, 45, 95, 133, 209, 247, 323, 361, 23, 20, 51, 115, 161, 253, 299, 391, 437, 529, 29, 22, 57, 125, 203, 319, 377, 493, 551, 667

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

Antti Karttunen, Table of n, a(n) for n = 2..3487; the first 83 antidiagonals of the array, flattened

Index entries for sequences that are permutations of the natural numbers

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.

Column 1: A000040, Column 2: A001248.

Row 1: A005843, Row 2: A016945, Row 3: A084967, Row 4: A084968, Row 5: A084969, Row 6: A084970.

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.

Also used in the definition of the following arrays of permutations: A249821, A251721, A251722.

Cf. A002260, A004736, A004280, A020639, A038179, A055396, A078898, A138511, A249820, A249730, A249735, A249744, A250469, A250470, A250472, A250474.

Sequence in context: A293054 A359298 A255127 * A246278 A359299 A363473

Adjacent sequences: A083218 A083219 A083220 * A083222 A083223 A083224

Keyword

nonn,tabl,look

Author

Yasutoshi Kohmoto, Jun 05 2003

Extensions

More terms from Hugo Pfoertner, Jun 13 2003

Status

approved