A286244 Square array A(n,k) = P(A046523(k), floor((n+k-1)/k)), 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, 3, 4, 10, 3, 5, 7, 3, 10, 3, 5, 11, 21, 3, 10, 5, 8, 16, 3, 21, 3, 10, 5, 8, 22, 36, 3, 21, 3, 14, 5, 12, 29, 10, 36, 3, 21, 3, 14, 8, 12, 37, 21, 10, 36, 3, 21, 5, 14, 8, 17, 46, 3, 21, 10, 36, 3, 21, 5, 14, 8, 17, 56, 78, 3, 21, 10, 36, 3, 27, 5, 19, 12, 23, 67, 3, 78, 3, 21, 10, 36, 3, 27, 5, 19, 12, 23, 79

Offset: 1

Author

Antti Karttunen, May 06 2017

Keywords

• nonn
• tabl

Comments

Transpose of A286245.

Examples

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

Links

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

Crossrefs

Transpose: A286245.

Cf. A000027, A046523, A286156, A286246, A286234.

Programs

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 T(a046523(k), int((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

Scheme

(define (A286244 n) (A286244bi (A002260 n) (A004736 n)))
(define (A286244bi row col) (let ((a (A046523 col)) (b (quotient (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2))))