A286236 Square array A(n,k) = P(A000010(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, 1, 2, 3, 0, 4, 3, 0, 2, 7, 10, 0, 0, 0, 11, 3, 0, 0, 5, 4, 16, 21, 0, 0, 0, 0, 0, 22, 10, 0, 0, 0, 5, 0, 7, 29, 21, 0, 0, 0, 0, 0, 8, 0, 37, 10, 0, 0, 0, 0, 14, 0, 0, 11, 46, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 10, 0, 0, 0, 0, 0, 5, 0, 8, 12, 16, 67, 78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79, 21, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 22, 92, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 17, 0, 106

Offset: 1

Author

Antti Karttunen, May 05 2017

Keywords

• nonn
• tabl

Comments

This is transpose of A286237, see comments there.

Examples

			The top left 12 X 12 corner of the array:
   1,  1,  3,  3, 10, 3, 21, 10, 21, 10, 55, 10
   2,  0,  0,  0,  0, 0,  0,  0,  0,  0,  0,  0
   4,  2,  0,  0,  0, 0,  0,  0,  0,  0,  0,  0
   7,  0,  5,  0,  0, 0,  0,  0,  0,  0,  0,  0
  11,  4,  0,  5,  0, 0,  0,  0,  0,  0,  0,  0
  16,  0,  0,  0, 14, 0,  0,  0,  0,  0,  0,  0
  22,  7,  8,  0,  0, 5,  0,  0,  0,  0,  0,  0
  29,  0,  0,  0,  0, 0, 27,  0,  0,  0,  0,  0
  37, 11,  0,  8,  0, 0,  0, 14,  0,  0,  0,  0
  46,  0, 12,  0,  0, 0,  0,  0, 27,  0,  0,  0
  56, 16,  0,  0, 19, 0,  0,  0,  0, 14,  0,  0
  67,  0,  0,  0,  0, 0,  0,  0,  0,  0, 65,  0
The first 15 rows when viewed as a triangle:
   1,
   1, 2,
   3, 0, 4,
   3, 0, 2, 7,
  10, 0, 0, 0, 11,
   3, 0, 0, 5,  4, 16,
  21, 0, 0, 0,  0,  0, 22,
  10, 0, 0, 0,  5,  0,  7, 29,
  21, 0, 0, 0,  0,  0,  8,  0, 37,
  10, 0, 0, 0,  0, 14,  0,  0, 11, 46,
  55, 0, 0, 0,  0,  0,  0,  0,  0,  0, 56,
  10, 0, 0, 0,  0,  0,  5,  0,  8, 12, 16, 67,
  78, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0, 79,
  21, 0, 0, 0,  0,  0,  0, 27,  0,  0,  0,  0, 22, 92,
  36, 0, 0, 0,  0,  0,  0,  0,  0,  0, 19,  0, 17,  0, 106
		

Links

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

Crossrefs

Transpose: A286237.

Cf. A000010, A000027, A113998, A286156, A286234, A286246.

Programs

Mathematica

T[n_, m_] := ((n + m)^2 - n - 3*m + 2)/2
t[n_, k_] := If[Mod[n, k] != 0, 0, T[EulerPhi[k], n/k]]
Table[Reverse[t[n, #] & /@ Range[n]], {n, 1, 20}] (* David Radcliffe, Jun 12 2025 *)

Python

from sympy import totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def t(n, k): return 0 if n%k!=0 else T(totient(k), n//k)
for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)][::-1]) # Indranil Ghosh, May 10 2017

Scheme

(define (A286236 n) (A286236bi (A002260 n) (A004736 n)))
(define (A286236bi row col) (if (not (zero? (modulo (+ row col -1) col))) 0 (let ((a (A000010 col)) (b (/ (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Alternatively, with triangular indexing:
(define (A286236 n) (A286236tr (A002024 n) (A002260 n)))
(define (A286236tr n k) (A286236bi k (+ 1 (- n k))))

Formula

T(n,k) = A113998(n,k) * A286234(n,k).