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A286245
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.
4
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, 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, 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
OFFSET
1,2
COMMENTS
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.
FORMULA
As a triangle (with n >= 1, 1 <= k <= n):
T(n,k) = (1/2)*(2 + ((A046523(k)+floor(n/k))^2) - A046523(k) - 3*floor(n/k)).
EXAMPLE
The first fifteen rows of triangle:
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, 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,
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,
92, 30, 12, 19, 5, 27, 5, 36, 10, 21, 3, 78, 3, 21,
106, 30, 17, 19, 8, 27, 5, 36, 10, 21, 3, 78, 3, 21, 21
PROG
(Scheme)
(define (A286245 n) (A286245bi (A002260 n) (A004736 n)))
(define (A286245bi row col) (let ((a (A046523 row)) (b (quotient (+ row col -1) row))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2))))
(Python)
from sympy import factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
f = factorint(n)
return sorted([f[i] for i in f])
def a046523(n):
x=1
while True:
if P(n) == P(x): return x
else: x+=1
def t(n, k): return T(a046523(k), int(n/k))
for n in range(1, 21): print [t(n, k) for k in range(1, n + 1)] # Indranil Ghosh, May 09 2017
CROSSREFS
Transpose: A286244.
Cf. A286247 (same triangle but with zeros in positions where k does not divide n), A286235.
Sequence in context: A242294 A234347 A328314 * A279849 A106826 A259582
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, May 06 2017
STATUS
approved