OFFSET
1,2
COMMENTS
This sequence packs the values of A046523(n/k) and k (whenever k divides n) to a single term with the pairing function A000027. The two "components" can be accessed with functions A002260 & A004736, which allows us to generate from this sequence (among other things) various sums related to the enumeration of aperiodic necklaces, because Moebius mu (A008683) obtains the same value on any representative of the same prime signature.
For example, we have:
and
Triangle A286247 has the same property.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 rows of triangle/antidiagonals of array
Eric Weisstein's World of Mathematics, Pairing Function
FORMULA
EXAMPLE
The first fifteen rows of triangle:
1,
3, 2,
3, 0, 4,
10, 5, 0, 7,
3, 0, 0, 0, 11,
21, 5, 8, 0, 0, 16,
3, 0, 0, 0, 0, 0, 22,
36, 14, 0, 12, 0, 0, 0, 29,
10, 0, 8, 0, 0, 0, 0, 0, 37,
21, 5, 0, 0, 17, 0, 0, 0, 0, 46,
3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56,
78, 27, 19, 12, 0, 23, 0, 0, 0, 0, 0, 67,
3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79,
21, 5, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 92,
21, 0, 8, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 106
-------------------------------------------------------------
Note how triangle A286247 contains on each row the same numbers in the same "divisibility-allotted" positions, but in reverse order.
PROG
(Scheme)
(define (A286249tr n k) (if (not (zero? (modulo n k))) 0 (let ((a (A046523 (/ n k))) (b k)) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
(Python)
from sympy import factorint
import math
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 0 if n%k!=0 else T(a046523(n/k), k)
for n in range(1, 21): print [t(n, k) for k in range(1, n + 1)] # Indranil Ghosh, May 08 2017
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, May 06 2017
STATUS
approved