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%I #42 Apr 19 2016 01:23:08
%S 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,
%T 143,169,17,16,39,85,119,187,221,289,19,18,45,95,133,209,247,323,361,
%U 23,20,51,115,161,253,299,391,437,529,29,22,57,125,203,319,377,493,551,667
%N 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), ...
%C This is permutation of natural numbers larger than 1.
%C From _Antti Karttunen_, Dec 19 2014: (Start)
%C 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.
%C For navigating in this array:
%C A055396(n) gives the row number of row where n occurs, and A078898(n) gives its column number, both starting their indexing from 1.
%C 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.
%C 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).
%C (End)
%C 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
%H Antti Karttunen, <a href="/A083221/b083221.txt">Table of n, a(n) for n = 2..3487; the first 83 antidiagonals of the array, flattened</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e The top left corner of the array:
%e 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26
%e 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75
%e 5, 25, 35, 55, 65, 85, 95, 115, 125, 145, 155, 175, 185
%e 7, 49, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329
%e 11, 121, 143, 187, 209, 253, 319, 341, 407, 451, 473, 517, 583
%e 13, 169, 221, 247, 299, 377, 403, 481, 533, 559, 611, 689, 767
%e 17, 289, 323, 391, 493, 527, 629, 697, 731, 799, 901, 1003, 1037
%e 19, 361, 437, 551, 589, 703, 779, 817, 893, 1007, 1121, 1159, 1273
%e 23, 529, 667, 713, 851, 943, 989, 1081, 1219, 1357, 1403, 1541, 1633
%e 29, 841, 899, 1073, 1189, 1247, 1363, 1537, 1711, 1769, 1943, 2059, 2117
%e ...
%t 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 *)
%o (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o (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.
%o (define (A083221bi row col) ((rowfun_n_for_A083221 row) col))
%o (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))))))))
%o (definec (A000040 n) ((rowfun_n_for_Esieve n) 1))
%o (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)))))))
%o ;; _Antti Karttunen_, Dec 19 2014
%Y Transpose of A083140.
%Y One more than A249741.
%Y Inverse permutation: A252460.
%Y Column 1: A000040, Column 2: A001248.
%Y Row 1: A005843, Row 2: A016945, Row 3: A084967, Row 4: A084968, Row 5: A084969, Row 6: A084970.
%Y Main diagonal: A083141.
%Y First semiprime in each column occurs at A251717; A251718 & A251719 with additional criteria. A251724 gives the corresponding semiprimes for the latter. See also A251728.
%Y Permutations based on mapping numbers between this array and A246278: A249817, A249818, A250244, A250245, A250247, A250249. See also: A249811, A249814, A249815.
%Y Also used in the definition of the following arrays of permutations: A249821, A251721, A251722.
%Y Cf. A002260, A004736, A004280, A020639, A038179, A055396, A078898, A138511, A249820, A249730, A249735, A249744, A250469, A250470, A250472, A250474.
%K nonn,tabl,look
%O 2,1
%A _Yasutoshi Kohmoto_, Jun 05 2003
%E More terms from _Hugo Pfoertner_, Jun 13 2003
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