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:
and
Triangle A286237 has the same property.
LINKS
FORMULA
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 (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
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, May 06 2017
STATUS
approved