A286235 Triangular table T(n,k) = P(phi(k), floor(n/k)), where P is sequence A000027 used as a pairing function N x N -> N, and phi is Euler totient function, A000010. Table is read by rows as T(1,1), T(2,1), T(2,2), etc.

1, 2, 1, 4, 1, 3, 7, 2, 3, 3, 11, 2, 3, 3, 10, 16, 4, 5, 3, 10, 3, 22, 4, 5, 3, 10, 3, 21, 29, 7, 5, 5, 10, 3, 21, 10, 37, 7, 8, 5, 10, 3, 21, 10, 21, 46, 11, 8, 5, 14, 3, 21, 10, 21, 10, 56, 11, 8, 5, 14, 3, 21, 10, 21, 10, 55, 67, 16, 12, 8, 14, 5, 21, 10, 21, 10, 55, 10, 79, 16, 12, 8, 14, 5, 21, 10, 21, 10, 55, 10, 78, 92, 22, 12, 8, 14, 5, 27, 10, 21, 10, 55, 10, 78, 21

Offset: 1

Author

Antti Karttunen, May 05 2017

Keywords

• nonn
• tabl

Comments

Equally: square array A(n,k) = P(A000010(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 the triangle:
    1,
    2,  1,
    4,  1,  3,
    7,  2,  3, 3,
   11,  2,  3, 3, 10,
   16,  4,  5, 3, 10, 3,
   22,  4,  5, 3, 10, 3, 21,
   29,  7,  5, 5, 10, 3, 21, 10,
   37,  7,  8, 5, 10, 3, 21, 10, 21,
   46, 11,  8, 5, 14, 3, 21, 10, 21, 10,
   56, 11,  8, 5, 14, 3, 21, 10, 21, 10, 55,
   67, 16, 12, 8, 14, 5, 21, 10, 21, 10, 55, 10,
   79, 16, 12, 8, 14, 5, 21, 10, 21, 10, 55, 10, 78,
   92, 22, 12, 8, 14, 5, 27, 10, 21, 10, 55, 10, 78, 21,
  106, 22, 17, 8, 19, 5, 27, 10, 21, 10, 55, 10, 78, 21, 36
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array
• Eric Weisstein's World of Mathematics, Pairing Function

Crossrefs

Transpose: A286234.

Cf. A000010, A000027, A286156, A286245.

Cf. A286237 (same triangle but with zeros in positions where k does not divide n).

Programs

Mathematica

Map[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ # & /@ # &, Table[{EulerPhi@ k, Floor[n/k]}, {n, 14}, {k, n}]] // Flatten (* Michael De Vlieger, May 06 2017 *)

Python

from sympy import totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def t(n, k): return T(totient(k), int(n//k))
for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 11 2017

Scheme

(define (A286235 n) (A286235bi (A002260 n) (A004736 n)))
(define (A286235bi row col) (let ((a (A000010 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 + ((A000010(k)+floor(n/k))^2) - A000010(k) - 3*floor(n/k)).