A286246 Square array A(n,k) = P(A046523(k), (n+k-1)/k) if k divides (n+k-1), 0 otherwise, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is a two-argument form of sequence A000027 used as a pairing function N x N -> N.

1, 3, 2, 3, 0, 4, 10, 0, 5, 7, 3, 0, 0, 0, 11, 21, 0, 0, 5, 8, 16, 3, 0, 0, 0, 0, 0, 22, 36, 0, 0, 0, 14, 0, 12, 29, 10, 0, 0, 0, 0, 0, 8, 0, 37, 21, 0, 0, 0, 0, 5, 0, 0, 17, 46, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 78, 0, 0, 0, 0, 0, 27, 0, 19, 12, 23, 67, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79, 21, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 30, 92, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 17, 0, 106

Offset

1,2

Links

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

Eric Weisstein's World of Mathematics, Pairing Function

Formula

T(n,k) = A113998(n,k) * A286244(n,k).

Example

The top left 12 X 12 corner of the array:
   1,  3,  3, 10,  3, 21,  3, 36, 10, 21,  3, 78
   2,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   4,  5,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   7,  0,  5,  0,  0,  0,  0,  0,  0,  0,  0,  0
  11,  8,  0, 14,  0,  0,  0,  0,  0,  0,  0,  0
  16,  0,  0,  0,  5,  0,  0,  0,  0,  0,  0,  0
  22, 12,  8,  0,  0, 27,  0,  0,  0,  0,  0,  0
  29,  0,  0,  0,  0,  0,  5,  0,  0,  0,  0,  0
  37, 17,  0, 19,  0,  0,  0, 44,  0,  0,  0,  0
  46,  0, 12,  0,  0,  0,  0,  0, 14,  0,  0,  0
  56, 23,  0,  0,  8,  0,  0,  0,  0, 27,  0,  0
  67,  0,  0,  0,  0,  0,  0,  0,  0,  0,  5,  0
The first fifteen rows of triangle:
   1,
   3, 2,
   3, 0, 4,
  10, 0, 5, 7,
   3, 0, 0, 0, 11,
  21, 0, 0, 5,  8, 16,
   3, 0, 0, 0,  0,  0, 22,
  36, 0, 0, 0, 14,  0, 12, 29,
  10, 0, 0, 0,  0,  0,  8,  0, 37,
  21, 0, 0, 0,  0,  5,  0,  0, 17, 46,
   3, 0, 0, 0,  0,  0,  0,  0,  0,  0, 56,
  78, 0, 0, 0,  0,  0, 27,  0, 19, 12, 23, 67,
   3, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0, 79,
  21, 0, 0, 0,  0,  0,  0,  5,  0,  0,  0,  0, 30, 92,
  21, 0, 0, 0,  0,  0,  0,  0,  0,  0,  8,  0, 17,  0, 106

Prog

(Scheme)
(define (A286246 n) (A286246bi (A002260 n) (A004736 n)))
(define (A286246bi row col) (if (not (zero? (modulo (+ row col -1) col))) 0 (let ((a (A046523 col)) (b (/ (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Alternatively, with triangular indexing:
(define (A286246 n) (A286246tr (A002024 n) (A002260 n)))
(define (A286246tr n k) (A286246bi k (+ 1 (- n k))))
(Python)
from sympy import factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def A(n, k): return 0 if (n + k - 1)%k!=0 else T(a046523(k), (n + k - 1)/k)
for n in range(1, 21): print [A(k, n - k + 1) for k in range(1, n + 1)] # Indranil Ghosh, May 09 2017

Crossrefs

Transpose: A286247.

Cf. A000027, A046523, A113998, A286156, A286244, A286236.

Sequence in context: A244543 A283979 A134676 * A286249 A239146 A324052

Adjacent sequences: A286243 A286244 A286245 * A286247 A286248 A286249

Keyword

nonn,tabl

Author

Antti Karttunen, May 06 2017

Status

approved