A286234 Square array A(n,k) = P(A000010(k), floor((n+k-1)/k)), 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, 1, 4, 3, 3, 2, 7, 10, 3, 3, 2, 11, 3, 10, 3, 5, 4, 16, 21, 3, 10, 3, 5, 4, 22, 10, 21, 3, 10, 5, 5, 7, 29, 21, 10, 21, 3, 10, 5, 8, 7, 37, 10, 21, 10, 21, 3, 14, 5, 8, 11, 46, 55, 10, 21, 10, 21, 3, 14, 5, 8, 11, 56, 10, 55, 10, 21, 10, 21, 5, 14, 8, 12, 16, 67, 78, 10, 55, 10, 21, 10, 21, 5, 14, 8, 12, 16, 79, 21, 78, 10, 55, 10, 21, 10, 27, 5, 14, 8, 12, 22, 92
Offset: 1
Examples
The top left 12 X 12 corner of the array: 1, 1, 3, 3, 10, 3, 21, 10, 21, 10, 55, 10 2, 1, 3, 3, 10, 3, 21, 10, 21, 10, 55, 10 4, 2, 3, 3, 10, 3, 21, 10, 21, 10, 55, 10 7, 2, 5, 3, 10, 3, 21, 10, 21, 10, 55, 10 11, 4, 5, 5, 10, 3, 21, 10, 21, 10, 55, 10 16, 4, 5, 5, 14, 3, 21, 10, 21, 10, 55, 10 22, 7, 8, 5, 14, 5, 21, 10, 21, 10, 55, 10 29, 7, 8, 5, 14, 5, 27, 10, 21, 10, 55, 10 37, 11, 8, 8, 14, 5, 27, 14, 21, 10, 55, 10 46, 11, 12, 8, 14, 5, 27, 14, 27, 10, 55, 10 56, 16, 12, 8, 19, 5, 27, 14, 27, 14, 55, 10 67, 16, 12, 8, 19, 5, 27, 14, 27, 14, 65, 10 The first fifteen rows when viewed as a triangle: 1 1 2 3 1 4 3 3 2 7 10 3 3 2 11 3 10 3 5 4 16 21 3 10 3 5 4 22 10 21 3 10 5 5 7 29 21 10 21 3 10 5 8 7 37 10 21 10 21 3 14 5 8 11 46 55 10 21 10 21 3 14 5 8 11 56 10 55 10 21 10 21 5 14 8 12 16 67 78 10 55 10 21 10 21 5 14 8 12 16 79 21 78 10 55 10 21 10 27 5 14 8 12 22 92 36 21 78 10 55 10 21 10 27 5 19 8 17 22 106
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
Programs
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Mathematica
Map[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ # & /@ Reverse@ # &, 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), n//k) for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)][::-1]) # Indranil Ghosh, May 11 2017
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Scheme
(define (A286234 n) (A286234bi (A002260 n) (A004736 n))) (define (A286234bi row col) (let ((a (A000010 col)) (b (quotient (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2))))
Comments