A285724 Square array read by descending antidiagonals: If n > k, A(n,k) = T(lcm(n,k), gcd(n,k)), otherwise A(n,k) = T(gcd(n,k), lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

1, 2, 3, 4, 5, 6, 7, 16, 21, 10, 11, 12, 13, 14, 15, 16, 46, 67, 78, 55, 21, 22, 23, 106, 25, 120, 27, 28, 29, 92, 31, 191, 210, 34, 105, 36, 37, 38, 211, 80, 41, 90, 231, 44, 45, 46, 154, 277, 379, 436, 465, 406, 300, 171, 55, 56, 57, 58, 59, 596, 61, 630, 63, 64, 65, 66, 67, 232, 436, 631, 781, 862, 903, 820, 666, 465, 253, 78, 79, 80, 529, 212, 991, 302, 85, 324, 1035, 230, 561, 90, 91

Offset: 1

Author

Antti Karttunen, May 03 2017

Keywords

• nonn
• tabl

Comments

The array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Examples

			The top left 12 X 12 corner of the array:
   1,   2,   4,   7,   11,   16,   22,   29,   37,   46,   56,   67
   3,   5,  16,  12,   46,   23,   92,   38,  154,   57,  232,   80
   6,  21,  13,  67,  106,   31,  211,  277,   58,  436,  529,   94
  10,  14,  78,  25,  191,   80,  379,   59,  631,  212,  947,  109
  15,  55, 120, 210,   41,  436,  596,  781,  991,   96, 1486, 1771
  21,  27,  34,  90,  465,   61,  862,  302,  193,  467, 2146,  142
  28, 105, 231, 406,  630,  903,   85, 1541, 1954, 2416, 2927, 3487
  36,  44, 300,  63,  820,  324, 1596,  113, 2557,  822, 3829,  355
  45, 171,  64, 666, 1035,  208, 2016, 2628,  145, 4006, 4852,  706
  55,  65, 465, 230,  101,  495, 2485,  860, 4095,  181, 5996, 1832
  66, 253, 561, 990, 1540, 2211, 3003, 3916, 4950, 6105,  221, 8647
  78,  90, 103, 117, 1830,  148, 3570,  375,  739, 1890, 8778,  265
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array
• MathWorld, Pairing Function

Crossrefs

Cf. A000124 (row 1), A000217 (column 1), A001844 (main diagonal).

Cf. A000027, A003989, A003990, A003991, A286101, A286102, A286155, A285722.

Programs

Scheme

(define (A285724 n) (A285724bi (A002260 n) (A004736 n)))
(define (A285724bi row col) (if (> row col) (A000027bi (lcm row col) (gcd row col)) (A000027bi (gcd row col) (lcm row col))))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

Formula

If n > k, A(n,k) = T(lcm(n,k),gcd(n,k)), otherwise A(n,k) = T(gcd(n,k),lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table, that is, as a pairing function from N x N to N.

If n < k, A(n,k) = A286101(n,k), otherwise A(n,k) = A286102(n,k).