A286249 Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(A046523(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.

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

Offset: 1

Author

Antti Karttunen, May 06 2017

Keywords

• nonn
• tabl

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:

Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * mu(A002260(a(i))) * 2^(A004736(a(i))) = A027375(n).

and

Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * mu(A002260(a(i))) * 3^(A004736(a(i))) = A054718(n).

Triangle A286247 has the same property.

Examples

			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.
		

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

Crossrefs

Transpose: A286248 (triangle reversed).

Cf. A000027, A008683, A027375, A046523, A054718, A054525, A286156, A286247, A286239.

Cf. A000124 (the right edge of the triangle).

Programs

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

Scheme

(define (A286249 n) (A286249tr (A002024 n) (A002260 n)))
(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)))))

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 + ((A046523(n/k)+k)^2) - A046523(n/k) - 3*k).