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

1,2

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:

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.

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

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).

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 (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)))))
(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

Crossrefs

Transpose: A286248 (triangle reversed).

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

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

Sequence in context: A283979 A134676 A286246 * A239146 A324052 A103491

Adjacent sequences: A286246 A286247 A286248 * A286250 A286251 A286252

Keyword

nonn,tabl

Author

Antti Karttunen, May 06 2017

Status

approved