A082005 -id:A082005 - cp's OEIS frontend

A082002 Square array read by antidiagonals, alternating upwards and downwards: T(1, 1) = 1 and every other entry is the smallest number not already used that has a common factor with every number in its row and column, not including the 1.

1, 2, 3, 6, 12, 4, 8, 18, 10, 9, 15, 24, 14, 30, 16, 20, 36, 22, 42, 40, 21, 27, 28, 48, 54, 26, 45, 32, 34, 60, 50, 66, 70, 56, 72, 33, 39, 44, 78, 84, 80, 75, 38, 51, 46, 52, 69, 68, 90, 96, 98, 102, 88, 104, 57, 63, 76, 108, 110, 114, 105, 100, 120, 92, 117, 58, 62, 87, 126

Offset: 1

Amarnath Murthy, Apr 05 2003

This is the boustrophedon method of filling an array.

			1,  2,  4,  8, 14, 16, ...
3, 12, 18, 24, ...
6, 10, 20, ...
9, 30, ...
15 ...

Cf. A082003, A082004, A082005.

More terms from David Wasserman, Jul 30 2004

A082003 First row of array in A082002.

Original entry on oeis.org

1, 2, 4, 8, 16, 20, 32, 34, 46, 52, 58, 62, 64, 74, 94, 106, 122, 128, 134, 136, 146, 158, 166, 178, 184, 194, 206, 214, 218, 226, 232, 254, 256, 262, 278, 298, 302, 314, 326, 334, 346, 358, 362, 376, 382, 386, 394, 398, 422, 424, 446, 454, 458, 464, 466, 478

Offset: 1

Amarnath Murthy, Apr 05 2003

nonn

Cf. A082002, A082004, A082005.

More terms from David Wasserman, Jul 30 2004

A082004 First column of array in A082002.

Original entry on oeis.org

1, 3, 6, 9, 15, 21, 27, 33, 39, 57, 63, 81, 93, 99, 111, 123, 129, 147, 171, 177, 189, 213, 231, 243, 267, 279, 291, 297, 303, 333, 339, 363, 369, 387, 393, 399, 405, 411, 441, 501, 513, 531, 537, 567, 573, 579, 591, 597, 627, 639, 651, 675, 681, 693, 699, 717

Offset: 1

Amarnath Murthy, Apr 05 2003

nonn

Cf. A082002, A082003, A082005.

More terms from David Wasserman, Jul 30 2004

A082017 First row of square array T(n,k) with T(1,1) = 1 where antidiagonals are filled alternating upwards and downwards with the smallest number not already used such that the n-th antidiagonal sum is a multiple of n.

Original entry on oeis.org

1, 2, 7, 6, 18, 15, 28, 29, 49, 45, 75, 66, 94, 91, 130, 120, 155, 153, 201, 190, 232, 231, 288, 276, 325, 326, 391, 378, 462, 435, 508, 496, 589, 561, 641, 630, 732, 703, 790, 780, 891, 861, 955, 946, 1066, 1035, 1136, 1128, 1257, 1225, 1333, 1326, 1464, 1431

Offset: 1

Amarnath Murthy, Apr 05 2003

nonn

			T(n,k) begins:
1,   2,  7,  6, 18, 15, ...
4,   5,  8, 14, 16, 27, ...
3,   9, 12, 17, 26, 31, ...
13, 11, 19, 25, 32, 42, ...
10, 20, 24, 33, 41, 50, ...
21, 23, 34, 39, 51, 60, ...

Cf. A074135, A082018, A082019, A082020, A082002, A082003, A082004, A082005.

Edited and more terms from Alois P. Heinz, Oct 26 2011

A082018 First column of square array T(n,k) with T(1,1) = 1 where antidiagonals are filled alternating upwards and downwards with the smallest number not already used such that the n-th antidiagonal sum is a multiple of n.

Original entry on oeis.org

1, 4, 3, 13, 10, 21, 22, 40, 36, 64, 55, 81, 78, 115, 105, 138, 136, 182, 171, 211, 210, 265, 253, 300, 301, 364, 351, 433, 406, 477, 465, 556, 528, 606, 595, 695, 666, 751, 741, 850, 820, 912, 903, 1021, 990, 1089, 1081, 1208, 1176, 1282, 1275, 1411, 1378

Offset: 1

Amarnath Murthy, Apr 05 2003

nonn

This is the boustrophedon method of filling an array. Sums of antidiagonals of T are in A074132. Sums of antidiagonals of T divided by number of antidiagonals are in A074133. Diagonal of T is in A082019.

			T(n,k) begins:
1,   2,  7,  6, 18, 15, ...
4,   5,  8, 14, 16, 27, ...
3,   9, 12, 17, 26, 31, ...
13, 11, 19, 25, 32, 42, ...
10, 20, 24, 33, 41, 50, ...
21, 23, 34, 39, 51, 60, ...

Cf. A074135, A082017, A082019, A082020, A082002, A082003, A082004, A082005.

Edited and more terms from Alois P. Heinz, Oct 26 2011

A082019 Diagonal of square array T(n,k) with T(1,1) = 1 where antidiagonals are filled alternating upwards and downwards with the smallest number not already used such that the n-th antidiagonal sum is a multiple of n.

Original entry on oeis.org

1, 5, 12, 25, 41, 60, 85, 112, 145, 180, 221, 264, 313, 365, 420, 481, 544, 613, 684, 761, 840, 925, 1012, 1105, 1200, 1301, 1404, 1513, 1624, 1741, 1860, 1985, 2112, 2245, 2380, 2521, 2664, 2813, 2964, 3121, 3281, 3444, 3613, 3784, 3961, 4140, 4325, 4512

Offset: 1

Amarnath Murthy, Apr 05 2003

nonn

			T(n,k) begins:
1,   2,  7,  6, 18, 15, ...
4,   5,  8, 14, 16, 27, ...
3,   9, 12, 17, 26, 31, ...
13, 11, 19, 25, 32, 42, ...
10, 20, 24, 33, 41, 50, ...
21, 23, 34, 39, 51, 60, ...

Cf. A074135, A082017, A082018, A082002, A082003, A082004, A082005.

Edited and more terms from Alois P. Heinz, Oct 26 2011

A082006 In the following square array numbers (not occurring earlier) are entered like this: a(1, 1), a(1, 2), a(2, 1), a(3, 1), a(2, 2), a(1, 3), a(1, 4), a(2, 3), a(3, 2), a(4, 1), a(5, 1), a(4, 2), ... such that every entry is coprime to the members of the row and column it belongs, with the condition that the n-th diagonal sum is a multiple of n. 1 2 7 9 31 25... 4 5 11 23 27... 3 13 8... 19 21... 17 ... ... Sequence contains numbers as they are entered.

Original entry on oeis.org

1, 2, 4, 3, 5, 7, 9, 11, 13, 19, 17, 21, 8, 23, 31, 25, 27, 29, 37, 41

Offset: 1

Amarnath Murthy, Apr 05 2003

Next term T(6,1) =a(21)> 500000, a(21) is odd. The sum of the first diagonal is 1 (a multiple of 1). The sum of the second diagonal is T(1,2)+T(2,1)=2+4=6 (a multiple of 2). The sum of the 3rd diagonal is T(1,3)+T(2,2)+T(3,1)=7+5+3=15 (a multiple of 3). The sum of the 4th diagonal is T(1,4)+T(2,3)+T(3,2)+T(4,1)=9+11+13+19=52 (a multiple of 4). The members of the first row (1,2,7,9,31,25,..) are mutually coprime. The members of the 2nd row (4,5,11,23,27,..) are mutually coprime. The members of the first column (1,4,3,19,17,..) are mutually coprime. The members of the 2nd column (2,5,13,21,..) are mutually coprime. The a(n) transverses the table in meandering fashion: first diagonal up, 2nd diagonal down, 3rd diagonal up, 4th down etc. - R. J. Mathar, May 06 2006
From Alois P. Heinz, Oct 06 2009: (Start)
T(6,1) is undefined, so there are no further terms.
For T(6,1) would be == 3 (mod 6) w.r.t. antidiagonal 6, (T(6,1)+159=6k) and it would be == 1 or == 5 (mod 6) w.r.t. column 1 (coprime to 3 & 4) which is impossible, unless backtracking is allowed and earlier elements are altered. But that is not intended by the author, because "sequence contains numbers as they are entered", and it would not make a valid definition at all. (End)

			Table is
1 2 7 9 31 25
4 5 11 23 27
3 13 8 29
19 21 37
17 41
?

Cf. A082002, A082003, A082004, A082005, A082007, A082008, A082009, A082010.

More terms from R. J. Mathar, May 06 2006

cp's OEIS Frontend

A082002 Square array read by antidiagonals, alternating upwards and downwards: T(1, 1) = 1 and every other entry is the smallest number not already used that has a common factor with every number in its row and column, not including the 1.

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A082003 First row of array in A082002.

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A082004 First column of array in A082002.

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A082017 First row of square array T(n,k) with T(1,1) = 1 where antidiagonals are filled alternating upwards and downwards with the smallest number not already used such that the n-th antidiagonal sum is a multiple of n.

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A082018 First column of square array T(n,k) with T(1,1) = 1 where antidiagonals are filled alternating upwards and downwards with the smallest number not already used such that the n-th antidiagonal sum is a multiple of n.

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A082019 Diagonal of square array T(n,k) with T(1,1) = 1 where antidiagonals are filled alternating upwards and downwards with the smallest number not already used such that the n-th antidiagonal sum is a multiple of n.

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Extensions

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Author

Keywords

Comments

Examples

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Extensions