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

1,2

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.

Links

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

MathWorld, Pairing Function

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)).

Example

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

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 *)

Prog

(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))))
(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

Crossrefs

Transpose: A286234.

Cf. A000010, A000027, A286156, A286245.

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

Sequence in context: A078072 A306944 A049776 * A180339 A079276 A210445

Adjacent sequences: A286232 A286233 A286234 * A286236 A286237 A286238

Keyword

nonn,tabl

Author

Antti Karttunen, May 05 2017

Status

approved