A286101 Square array A(n,k) read by antidiagonals: 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, 2, 4, 5, 4, 7, 16, 16, 7, 11, 12, 13, 12, 11, 16, 46, 67, 67, 46, 16, 22, 23, 106, 25, 106, 23, 22, 29, 92, 31, 191, 191, 31, 92, 29, 37, 38, 211, 80, 41, 80, 211, 38, 37, 46, 154, 277, 379, 436, 436, 379, 277, 154, 46, 56, 57, 58, 59, 596, 61, 596, 59, 58, 57, 56, 67, 232, 436, 631, 781, 862, 862, 781, 631, 436, 232, 67, 79, 80, 529, 212, 991, 302, 85, 302, 991, 212, 529, 80, 79

Offset

1,2

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.

Links

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

MathWorld, Pairing Function

Formula

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.

A(n,k) = A(k,n), or equivalently, a(A038722(n)) = a(n). [Array is symmetric.]

Example

The top left 12 X 12 corner of the array:
   1,   2,   4,   7,   11,   16,   22,   29,   37,   46,   56,   67
   2,   5,  16,  12,   46,   23,   92,   38,  154,   57,  232,   80
   4,  16,  13,  67,  106,   31,  211,  277,   58,  436,  529,   94
   7,  12,  67,  25,  191,   80,  379,   59,  631,  212,  947,  109
  11,  46, 106, 191,   41,  436,  596,  781,  991,   96, 1486, 1771
  16,  23,  31,  80,  436,   61,  862,  302,  193,  467, 2146,  142
  22,  92, 211, 379,  596,  862,   85, 1541, 1954, 2416, 2927, 3487
  29,  38, 277,  59,  781,  302, 1541,  113, 2557,  822, 3829,  355
  37, 154,  58, 631,  991,  193, 1954, 2557,  145, 4006, 4852,  706
  46,  57, 436, 212,   96,  467, 2416,  822, 4006,  181, 5996, 1832
  56, 232, 529, 947, 1486, 2146, 2927, 3829, 4852, 5996,  221, 8647
  67,  80,  94, 109, 1771,  142, 3487,  355,  706, 1832, 8647,  265

Prog

(Scheme)
(define (A286101 n) (A286101bi (A002260 n) (A004736 n)))
(define (A286101bi row col) (A000027bi (gcd row col) (lcm row col)))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

Crossrefs

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

Cf. A000027, A003989, A003990, A003991, A038722, A285722, A285732, A286098, A286099, A286102, A285724.

Sequence in context: A136099 A210597 A252836 * A072454 A115216 A208637

Adjacent sequences: A286098 A286099 A286100 * A286102 A286103 A286104

Keyword

nonn,tabl

Author

Antti Karttunen, May 03 2017

Status

approved