login
A286239
Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(A000010(n/k), k), where P is sequence A000027 used as a pairing function N x N -> N. Table is read by rows as T(1,1), T(2,1), T(2,2), etc.
4
1, 1, 2, 3, 0, 4, 3, 2, 0, 7, 10, 0, 0, 0, 11, 3, 5, 4, 0, 0, 16, 21, 0, 0, 0, 0, 0, 22, 10, 5, 0, 7, 0, 0, 0, 29, 21, 0, 8, 0, 0, 0, 0, 0, 37, 10, 14, 0, 0, 11, 0, 0, 0, 0, 46, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 10, 5, 8, 12, 0, 16, 0, 0, 0, 0, 0, 67, 78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79, 21, 27, 0, 0, 0, 0, 22, 0, 0, 0, 0, 0, 0, 92, 36, 0, 19, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 106
OFFSET
1,3
COMMENTS
This sequence packs the values of phi(n/k) and k (whenever k divides n) to a single value, with the pairing function A000027. The two "components" can be accessed with functions A002260 & A004736, which allows us generate from this sequence various sums related to necklace enumeration (among other things).
For example, we have:
Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * A002260(a(i)) * 2^(A004736(a(i))) = A053635(n).
and
Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * A002260(a(i)) * 3^(A004736(a(i))) = A054610(n)
Triangle A286237 has the same property.
FORMULA
As a triangle (with n >= 1, 1 <= k <= n):
T(n,k) = 0 if k does not divide n, otherwise T(n,k) = (1/2)*(2 + ((A000010(n/k)+k)^2) - A000010(n/k) - 3*k).
EXAMPLE
The first fifteen rows of triangle:
1,
1, 2,
3, 0, 4,
3, 2, 0, 7,
10, 0, 0, 0, 11,
3, 5, 4, 0, 0, 16,
21, 0, 0, 0, 0, 0, 22,
10, 5, 0, 7, 0, 0, 0, 29,
21, 0, 8, 0, 0, 0, 0, 0, 37,
10, 14, 0, 0, 11, 0, 0, 0, 0, 46,
55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56,
10, 5, 8, 12, 0, 16, 0, 0, 0, 0, 0, 67,
78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79,
21, 27, 0, 0, 0, 0, 22, 0, 0, 0, 0, 0, 0, 92,
36, 0, 19, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 106
-------------------------------------------------------------
Note how triangle A286237 contains on each row the same numbers in the same "divisibility-allotted" positions, but in reverse order.
PROG
(Scheme)
(define (A286239 n) (A286239tr (A002024 n) (A002260 n)))
(define (A286239tr n k) (if (not (zero? (modulo n k))) 0 (let ((a (A000010 (/ n k))) (b k)) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
(Python)
from sympy import totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def t(n, k): return 0 if n%k!=0 else T(totient(n/k), 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: A286238.
Cf. A000124 (the right edge of the triangle).
Sequence in context: A327800 A286236 A230451 * A341585 A343866 A140502
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, May 06 2017
STATUS
approved