A286245 Triangular table T(n,k) = P(A046523(k), floor(n/k)), read by rows as T(1,1), T(2,1), T(2,2), etc. Here P is sequence A000027 used as a pairing function N x N -> N.

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

Offset: 1

Author

Antti Karttunen, May 06 2017

Keywords

• nonn
• tabl

Comments

Equally: square array A(n,k) = P(A046523(n), floor((n+k-1)/n)), 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.

Examples

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

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

Crossrefs

Transpose: A286244.

Cf. A000027, A046523, A286156.

Cf. A286247 (same triangle but with zeros in positions where k does not divide n), A286235.

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 t(n, k): return T(a046523(k), int(n//k))
for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 09 2017

Scheme

(define (A286245 n) (A286245bi (A002260 n) (A004736 n)))
(define (A286245bi row col) (let ((a (A046523 row)) (b (quotient (+ row col -1) row))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2))))

Formula

As a triangle (with n >= 1, 1 <= k <= n):

T(n,k) = (1/2)*(2 + ((A046523(k)+floor(n/k))^2) - A046523(k) - 3*floor(n/k)).