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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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 xrange(1, 21): print [t(n, k) for k in xrange(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

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Last modified October 16 15:51 EDT 2019. Contains 328101 sequences. (Running on oeis4.)