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%I #21 Dec 07 2019 12:18:29
%S 1,2,3,4,3,3,7,5,3,10,11,5,3,10,3,16,8,5,10,3,21,22,8,5,10,3,21,3,29,
%T 12,5,14,3,21,3,36,37,12,8,14,3,21,3,36,10,46,17,8,14,5,21,3,36,10,21,
%U 56,17,8,14,5,21,3,36,10,21,3,67,23,12,19,5,27,3,36,10,21,3,78,79,23,12,19,5,27,3,36,10,21,3,78,3
%N Triangular table T(n,k) = P(A046523(k), floor(n/k)), read by rows as T(1,1), T(2,1), T(2,2), etc. Here P is sequence A000027 used as a pairing function N x N -> N.
%C Equally: square array A(n,k) = P(A046523(n), floor((n+k-1)/n)), 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="/A286245/b286245.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="http://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F As a triangle (with n >= 1, 1 <= k <= n):
%F T(n,k) = (1/2)*(2 + ((A046523(k)+floor(n/k))^2) - A046523(k) - 3*floor(n/k)).
%e The first fifteen rows of triangle:
%e 1,
%e 2, 3,
%e 4, 3, 3,
%e 7, 5, 3, 10,
%e 11, 5, 3, 10, 3,
%e 16, 8, 5, 10, 3, 21,
%e 22, 8, 5, 10, 3, 21, 3,
%e 29, 12, 5, 14, 3, 21, 3, 36,
%e 37, 12, 8, 14, 3, 21, 3, 36, 10,
%e 46, 17, 8, 14, 5, 21, 3, 36, 10, 21,
%e 56, 17, 8, 14, 5, 21, 3, 36, 10, 21, 3,
%e 67, 23, 12, 19, 5, 27, 3, 36, 10, 21, 3, 78,
%e 79, 23, 12, 19, 5, 27, 3, 36, 10, 21, 3, 78, 3,
%e 92, 30, 12, 19, 5, 27, 5, 36, 10, 21, 3, 78, 3, 21,
%e 106, 30, 17, 19, 8, 27, 5, 36, 10, 21, 3, 78, 3, 21, 21
%o (Scheme)
%o (define (A286245 n) (A286245bi (A002260 n) (A004736 n)))
%o (define (A286245bi row col) (let ((a (A046523 row)) (b (quotient (+ row col -1) row))) (* (/ 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 t(n, k): return T(a046523(k), int(n/k))
%o for n in range(1, 21): print [t(n, k) for k in range(1, n + 1)] # _Indranil Ghosh_, May 09 2017
%Y Transpose: A286244.
%Y Cf. A000027, A046523, A286156.
%Y Cf. A286247 (same triangle but with zeros in positions where k does not divide n), A286235.
%K nonn,tabl
%O 1,2
%A _Antti Karttunen_, May 06 2017
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