%I #19 Feb 16 2025 08:33:44
%S 1,3,2,3,0,4,10,0,5,7,3,0,0,0,11,21,0,0,5,8,16,3,0,0,0,0,0,22,36,0,0,
%T 0,14,0,12,29,10,0,0,0,0,0,8,0,37,21,0,0,0,0,5,0,0,17,46,3,0,0,0,0,0,
%U 0,0,0,0,56,78,0,0,0,0,0,27,0,19,12,23,67,3,0,0,0,0,0,0,0,0,0,0,0,79,21,0,0,0,0,0,0,5,0,0,0,0,30,92,21,0,0,0,0,0,0,0,0,0,8,0,17,0,106
%N Square array A(n,k) = P(A046523(k), (n+k-1)/k) if k divides (n+k-1), 0 otherwise, 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.
%H Antti Karttunen, <a href="/A286246/b286246.txt">Table of n, a(n) for n = 1..10585; the first 145 rows of triangle/antidiagonals of array</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F T(n,k) = A113998(n,k) * A286244(n,k).
%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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e 4, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e 7, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e 11, 8, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0
%e 16, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0
%e 22, 12, 8, 0, 0, 27, 0, 0, 0, 0, 0, 0
%e 29, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0
%e 37, 17, 0, 19, 0, 0, 0, 44, 0, 0, 0, 0
%e 46, 0, 12, 0, 0, 0, 0, 0, 14, 0, 0, 0
%e 56, 23, 0, 0, 8, 0, 0, 0, 0, 27, 0, 0
%e 67, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0
%e The first fifteen rows of triangle:
%e 1,
%e 3, 2,
%e 3, 0, 4,
%e 10, 0, 5, 7,
%e 3, 0, 0, 0, 11,
%e 21, 0, 0, 5, 8, 16,
%e 3, 0, 0, 0, 0, 0, 22,
%e 36, 0, 0, 0, 14, 0, 12, 29,
%e 10, 0, 0, 0, 0, 0, 8, 0, 37,
%e 21, 0, 0, 0, 0, 5, 0, 0, 17, 46,
%e 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56,
%e 78, 0, 0, 0, 0, 0, 27, 0, 19, 12, 23, 67,
%e 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79,
%e 21, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 30, 92,
%e 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 17, 0, 106
%o (Scheme)
%o (define (A286246 n) (A286246bi (A002260 n) (A004736 n)))
%o (define (A286246bi row col) (if (not (zero? (modulo (+ row col -1) col))) 0 (let ((a (A046523 col)) (b (/ (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
%o ;; Alternatively, with triangular indexing:
%o (define (A286246 n) (A286246tr (A002024 n) (A002260 n)))
%o (define (A286246tr n k) (A286246bi k (+ 1 (- n k))))
%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 0 if (n + k - 1)%k!=0 else T(a046523(k), (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: A286247.
%Y Cf. A000027, A046523, A113998, A286156, A286244, A286236.
%K nonn,tabl,changed
%O 1,2
%A _Antti Karttunen_, May 06 2017