%I #19 Feb 16 2025 08:33:44
%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="https://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,changed
%O 1,2
%A _Antti Karttunen_, May 06 2017