A286244 - OEIS

%I #18 Dec 07 2019 12:18:29

%S 1,3,2,3,3,4,10,3,5,7,3,10,3,5,11,21,3,10,5,8,16,3,21,3,10,5,8,22,36,

%T 3,21,3,14,5,12,29,10,36,3,21,3,14,8,12,37,21,10,36,3,21,5,14,8,17,46,

%U 3,21,10,36,3,21,5,14,8,17,56,78,3,21,10,36,3,27,5,19,12,23,67,3,78,3,21,10,36,3,27,5,19,12,23,79

%N Square array A(n,k) = P(A046523(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.

%C Transpose of A286245.

%H Antti Karttunen, <a href="/A286244/b286244.txt">Table of n, a(n) for n = 1..10585; the first 145 rows of triangle/antidiagonals of array</a>

%H MathWorld, <a href="http://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>

%e The top left 12 X 12 corner of the array:

%e    1,  3,  3, 10, 3, 21, 3, 36, 10, 21, 3, 78

%e    2,  3,  3, 10, 3, 21, 3, 36, 10, 21, 3, 78

%e    4,  5,  3, 10, 3, 21, 3, 36, 10, 21, 3, 78

%e    7,  5,  5, 10, 3, 21, 3, 36, 10, 21, 3, 78

%e   11,  8,  5, 14, 3, 21, 3, 36, 10, 21, 3, 78

%e   16,  8,  5, 14, 5, 21, 3, 36, 10, 21, 3, 78

%e   22, 12,  8, 14, 5, 27, 3, 36, 10, 21, 3, 78

%e   29, 12,  8, 14, 5, 27, 5, 36, 10, 21, 3, 78

%e   37, 17,  8, 19, 5, 27, 5, 44, 10, 21, 3, 78

%e   46, 17, 12, 19, 5, 27, 5, 44, 14, 21, 3, 78

%e   56, 23, 12, 19, 8, 27, 5, 44, 14, 27, 3, 78

%e   67, 23, 12, 19, 8, 27, 5, 44, 14, 27, 5, 78

%e The first fifteen rows when viewed as a triangle:

%e    1,

%e    3,  2,

%e    3,  3,  4,

%e   10,  3,  5,  7,

%e    3, 10,  3,  5, 11,

%e   21,  3, 10,  5,  8, 16,

%e    3, 21,  3, 10,  5,  8, 22,

%e   36,  3, 21,  3, 14,  5, 12, 29,

%e   10, 36,  3, 21,  3, 14,  8, 12, 37,

%e   21, 10, 36,  3, 21,  5, 14,  8, 17, 46,

%e    3, 21, 10, 36,  3, 21,  5, 14,  8, 17, 56,

%e   78,  3, 21, 10, 36,  3, 27,  5, 19, 12, 23, 67,

%e    3, 78,  3, 21, 10, 36,  3, 27,  5, 19, 12, 23, 79,

%e   21,  3, 78,  3, 21, 10, 36,  5, 27,  5, 19, 12, 30, 92,

%e   21, 21,  3, 78,  3, 21, 10, 36,  5, 27,  8, 19, 17, 30, 106

%o (Scheme)

%o (define (A286244 n) (A286244bi (A002260 n) (A004736 n)))

%o (define (A286244bi row col) (let ((a (A046523 col)) (b (quotient (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2))))

%o (Python)

%o from sympy import factorint

%o def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2

%o def P(n):

%o     f = factorint(n)

%o     return sorted([f[i] for i in f])

%o def a046523(n):

%o     x=1

%o     while True:

%o         if P(n) == P(x): return x

%o         else: x+=1

%o def A(n, k): return T(a046523(k), int((n + k - 1)/k))

%o for n in range(1, 21): print [A(k, n - k + 1) for k in range(1, n + 1)] # _Indranil Ghosh_, May 09 2017

%Y Transpose: A286245.

%Y Cf. A000027, A046523, A286156, A286246, A286234.

%K nonn,tabl

%O 1,2

%A _Antti Karttunen_, May 06 2017