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A286244 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.

%I A286244 #23 Jun 10 2025 13:53:37
%S A286244 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 A286244 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 A286244 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 A286244 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 A286244 Transpose of A286245.
%H A286244 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 A286244 MathWorld, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%e A286244 The top left 12 X 12 corner of the array:
%e A286244    1,  3,  3, 10, 3, 21, 3, 36, 10, 21, 3, 78
%e A286244    2,  3,  3, 10, 3, 21, 3, 36, 10, 21, 3, 78
%e A286244    4,  5,  3, 10, 3, 21, 3, 36, 10, 21, 3, 78
%e A286244    7,  5,  5, 10, 3, 21, 3, 36, 10, 21, 3, 78
%e A286244   11,  8,  5, 14, 3, 21, 3, 36, 10, 21, 3, 78
%e A286244   16,  8,  5, 14, 5, 21, 3, 36, 10, 21, 3, 78
%e A286244   22, 12,  8, 14, 5, 27, 3, 36, 10, 21, 3, 78
%e A286244   29, 12,  8, 14, 5, 27, 5, 36, 10, 21, 3, 78
%e A286244   37, 17,  8, 19, 5, 27, 5, 44, 10, 21, 3, 78
%e A286244   46, 17, 12, 19, 5, 27, 5, 44, 14, 21, 3, 78
%e A286244   56, 23, 12, 19, 8, 27, 5, 44, 14, 27, 3, 78
%e A286244   67, 23, 12, 19, 8, 27, 5, 44, 14, 27, 5, 78
%e A286244 The first fifteen rows when viewed as a triangle:
%e A286244    1,
%e A286244    3,  2,
%e A286244    3,  3,  4,
%e A286244   10,  3,  5,  7,
%e A286244    3, 10,  3,  5, 11,
%e A286244   21,  3, 10,  5,  8, 16,
%e A286244    3, 21,  3, 10,  5,  8, 22,
%e A286244   36,  3, 21,  3, 14,  5, 12, 29,
%e A286244   10, 36,  3, 21,  3, 14,  8, 12, 37,
%e A286244   21, 10, 36,  3, 21,  5, 14,  8, 17, 46,
%e A286244    3, 21, 10, 36,  3, 21,  5, 14,  8, 17, 56,
%e A286244   78,  3, 21, 10, 36,  3, 27,  5, 19, 12, 23, 67,
%e A286244    3, 78,  3, 21, 10, 36,  3, 27,  5, 19, 12, 23, 79,
%e A286244   21,  3, 78,  3, 21, 10, 36,  5, 27,  5, 19, 12, 30, 92,
%e A286244   21, 21,  3, 78,  3, 21, 10, 36,  5, 27,  8, 19, 17, 30, 106
%o A286244 (Scheme)
%o A286244 (define (A286244 n) (A286244bi (A002260 n) (A004736 n)))
%o A286244 (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 A286244 (Python)
%o A286244 from sympy import factorint
%o A286244 def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
%o A286244 def P(n):
%o A286244     f = factorint(n)
%o A286244     return sorted([f[i] for i in f])
%o A286244 def a046523(n):
%o A286244     x=1
%o A286244     while True:
%o A286244         if P(n) == P(x): return x
%o A286244         else: x+=1
%o A286244 def A(n, k): return T(a046523(k), int((n + k - 1)//k))
%o A286244 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 A286244 Transpose: A286245.
%Y A286244 Cf. A000027, A046523, A286156, A286246, A286234.
%K A286244 nonn,tabl
%O A286244 1,2
%A A286244 _Antti Karttunen_, May 06 2017