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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.
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%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