|
| |
|
|
A286237
|
|
Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(phi(k), n/k), where P is sequence A000027 used as a pairing function N x N -> N, and phi is Euler totient function, A000010. Table is read by rows as T(1,1), T(2,1), T(2,2), etc.
|
|
5
|
|
|
|
1, 2, 1, 4, 0, 3, 7, 2, 0, 3, 11, 0, 0, 0, 10, 16, 4, 5, 0, 0, 3, 22, 0, 0, 0, 0, 0, 21, 29, 7, 0, 5, 0, 0, 0, 10, 37, 0, 8, 0, 0, 0, 0, 0, 21, 46, 11, 0, 0, 14, 0, 0, 0, 0, 10, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 67, 16, 12, 8, 0, 5, 0, 0, 0, 0, 0, 10, 79, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78, 92, 22, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 21, 106, 0, 17, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Equally: square array A(n,k) = P(A000010(n), (n+k-1)/n) if n divides (n+k-1), 0 otherwise, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is sequence A000027 used as a pairing function N x N -> N.
When viewed as a triangular table, this sequence packs the values of phi(k) and quotient n/k (when it is integral) 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:
and
|
|
|
LINKS
|
|
|
|
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(k)+(n/k))^2) - A000010(k) - 3*(n/k)).
Other identities. For all n >= 1:
T(prime(n),prime(n)) = A000217(prime(n)-1).
|
|
|
EXAMPLE
|
The first fifteen rows of the triangle:
1,
2, 1,
4, 0, 3,
7, 2, 0, 3,
11, 0, 0, 0, 10,
16, 4, 5, 0, 0, 3,
22, 0, 0, 0, 0, 0, 21,
29, 7, 0, 5, 0, 0, 0, 10,
37, 0, 8, 0, 0, 0, 0, 0, 21,
46, 11, 0, 0, 14, 0, 0, 0, 0, 10,
56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55,
67, 16, 12, 8, 0, 5, 0, 0, 0, 0, 0, 10,
79, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78,
92, 22, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 21,
106, 0, 17, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36
---------------------------------------------------------------
Note how triangle A286239 contains on each row the same numbers in the same "divisibility-allotted" positions, but in reverse order.
In the following examples: a = this sequence interpreted as a one-dimensional sequence, A = interpreted as a square array, T = interpreted as a triangular table, P = A000027 interpreted as a pairing function N x N -> N, phi = Euler totient function, A000010.
---
a(7) = A(1,4) = T(4,1) = P(phi(1),4/1) = P(1,4) = 1+(((1+4)^2 - 1 - (3*4))/2) = 7.
a(30) = A(2,7) = T(8,2) = P(phi(2),8/2) = P(1,4) (i.e., same as above) = 7.
a(10) = A(5,1) = T(5,5) = P(phi(5),5/5) = P(4,1) = 1+(((4+1)^2 - 4 - (3*1))/2) = 10.
a(110) = A(5,11) = T(15,5) = P(phi(5),15/5) = P(4,3) = 1+((4+3)^2 - 4 - (3*3))/2 = 19.
|
|
|
PROG
|
(Scheme)
(define (A286237bi row col) (if (not (zero? (modulo (+ row col -1) row))) 0 (let ((a (A000010 row)) (b (quotient (+ row col -1) row))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Alternatively, with triangular indexing:
(define (A286237tr n k) (if (not (zero? (modulo n k))) 0 (let ((a (A000010 k)) (b (/ n k))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Note that: (A286237tr n k) is equal to (A286237bi k (+ 1 (- n k))).
(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(k), n/k)
for n in range(1, 21): print [t(n, k) for k in range(1, n + 1)] # Indranil Ghosh, May 10 2017
|
|
|
CROSSREFS
|
Cf. A000124 (left edge of the triangle), A000217 (every number at the right edge is a triangular number).
Cf. A000010, A000027, A002024, A002260, A004736, A051731, A053635, A054610, A286235, A286239, A286247.
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
