A083221 - OEIS

%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