A286247 Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(A046523(k), n/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, 2, 3, 4, 0, 3, 7, 5, 0, 10, 11, 0, 0, 0, 3, 16, 8, 5, 0, 0, 21, 22, 0, 0, 0, 0, 0, 3, 29, 12, 0, 14, 0, 0, 0, 36, 37, 0, 8, 0, 0, 0, 0, 0, 10, 46, 17, 0, 0, 5, 0, 0, 0, 0, 21, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 67, 23, 12, 19, 0, 27, 0, 0, 0, 0, 0, 78, 79, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 92, 30, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 21, 106, 0, 17, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21

Offset

1,2

Comments

Equally: square array A(n,k) = P(A046523(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 a two-argument form of sequence A000027 used as a pairing function N x N -> N.

When viewed as a triangular table, this sequence packs the values of A046523(k) [which essentially stores the prime signature of 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 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 A286249 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(k)+(n/k))^2) - A046523(k) - 3*(n/k)).

T(n,k) = A051731(n,k) * A286245(n,k).

Example

The first fifteen rows of triangle:
    1,
    2,  3,
    4,  0,  3,
    7,  5,  0, 10,
   11,  0,  0,  0,  3,
   16,  8,  5,  0,  0, 21,
   22,  0,  0,  0,  0,  0,  3,
   29, 12,  0, 14,  0,  0,  0, 36,
   37,  0,  8,  0,  0,  0,  0,  0, 10,
   46, 17,  0,  0,  5,  0,  0,  0,  0, 21,
   56,  0,  0,  0,  0,  0,  0,  0,  0,  0, 3,
   67, 23, 12, 19,  0, 27,  0,  0,  0,  0,  0, 78,
   79,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  3,
   92, 30,  0,  0,  0,  0,  5,  0,  0,  0,  0,  0,  0, 21,
  106,  0, 17,  0,  8,  0,  0,  0,  0,  0,  0,  0,  0,  0, 21
  (Note how triangle A286249 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, T = interpreted as a triangular table, A = interpreted as a square array, P = A000027 interpreted as a two-argument pairing function N x N -> N.
---
a(7) = T(4,1) = A(1,4) = P(A046523(1),4/1) = P(1,4) = 1+(((1+4)^2 - 1 - (3*4))/2) = 7.
a(30) = T(8,2) = A(2,7) = P(A046523(2),8/2) = P(2,4) = (1/2)*(2 + ((2+4)^2) - 2 - 3*4) = 12.

Prog

(Scheme)
(define (A286247 n) (A286247bi (A002260 n) (A004736 n)))
(define (A286247bi row col) (if (not (zero? (modulo (+ row col -1) row))) 0 (let ((a (A046523 row)) (b (quotient (+ row col -1) row))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Alternatively, with triangular indexing:
(define (A286247 n) (A286247tr (A002024 n) (A002260 n)))
(define (A286247tr n k) (if (not (zero? (modulo n k))) 0 (let ((a (A046523 k)) (b (/ n k))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
(Python)
from sympy import factorint
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(k), n/k)
for n in range(1, 21): print [t(n, k) for k in range(1, n + 1)] # Indranil Ghosh, May 08 2017

Crossrefs

Cf. A000027, A002024, A002260, A004736, A008683, A027375, A046523, A051731, A054718, A286245, A286249, A286237.

Cf. A000124 (left edge of the triangle), A000217 (every number at the right edge is a triangular number).

Sequence in context: A279125 A011025 A286248 * A203571 A199511 A049279

Adjacent sequences: A286244 A286245 A286246 * A286248 A286249 A286250

Keyword

nonn,tabl

Author

Antti Karttunen, May 06 2017

Status

approved