A286237 Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(phi(k), n/k), where P is sequence A000027 used as a pairing function N x N -> N, and phi is Euler totient function, A000010. Table is read by rows as T(1,1), T(2,1), T(2,2), etc.

1, 2, 1, 4, 0, 3, 7, 2, 0, 3, 11, 0, 0, 0, 10, 16, 4, 5, 0, 0, 3, 22, 0, 0, 0, 0, 0, 21, 29, 7, 0, 5, 0, 0, 0, 10, 37, 0, 8, 0, 0, 0, 0, 0, 21, 46, 11, 0, 0, 14, 0, 0, 0, 0, 10, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 67, 16, 12, 8, 0, 5, 0, 0, 0, 0, 0, 10, 79, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78, 92, 22, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 21, 106, 0, 17, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36

Offset: 1

Author

Antti Karttunen, May 05 2017

Keywords

• nonn
• tabl

Comments

Equally: square array A(n,k) = P(A000010(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 sequence A000027 used as a pairing function N x N -> N.

When viewed as a triangular table, this sequence packs the values of phi(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 generate from this sequence various sums related to necklace enumeration (among other things).

For example, we have:

Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * A002260(a(i)) * 2^(A004736(a(i))) = A053635(n).

and

Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * A002260(a(i)) * 3^(A004736(a(i))) = A054610(n).

Examples

			The first fifteen rows of the triangle:
    1,
    2,  1,
    4,  0,  3,
    7,  2,  0,  3,
   11,  0,  0,  0, 10,
   16,  4,  5,  0,  0,  3,
   22,  0,  0,  0,  0,  0, 21,
   29,  7,  0,  5,  0,  0,  0, 10,
   37,  0,  8,  0,  0,  0,  0,  0, 21,
   46, 11,  0,  0, 14,  0,  0,  0,  0, 10,
   56,  0,  0,  0,  0,  0,  0,  0,  0,  0, 55,
   67, 16, 12,  8,  0,  5,  0,  0,  0,  0,  0, 10,
   79,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 78,
   92, 22,  0,  0,  0,  0, 27,  0,  0,  0,  0,  0,  0, 21,
  106,  0, 17,  0, 19,  0,  0,  0,  0,  0,  0,  0,  0,  0, 36
---------------------------------------------------------------
Note how triangle A286239 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, A = interpreted as a square array, T = interpreted as a triangular table, P = A000027 interpreted as a pairing function N x N -> N, phi = Euler totient function, A000010.
---
a(7) = A(1,4) = T(4,1) = P(phi(1),4/1) = P(1,4) = 1+(((1+4)^2 - 1 - (3*4))/2) = 7.
a(30) = A(2,7) = T(8,2) = P(phi(2),8/2) = P(1,4) (i.e., same as above) = 7.
a(10) = A(5,1) = T(5,5) = P(phi(5),5/5) = P(4,1) = 1+(((4+1)^2 - 4 - (3*1))/2) = 10.
a(110) = A(5,11) = T(15,5) = P(phi(5),15/5) = P(4,3) = 1+((4+3)^2 - 4 - (3*3))/2 = 19.
		

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: A286236.

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

Cf. A000010, A000027, A002024, A002260, A004736, A051731, A053635, A054610, A286235, A286239, A286247.

Programs

Mathematica

(* Based on Python script by Indranil Ghosh *)
T[n_, m_] := ((n + m)^2 - n - 3*m + 2)/2
t[n_, k_] := If[Mod[n, k] != 0, 0, T[EulerPhi[k], n/k]]
Table[t[n, k], {n, 1, 20}, {k, 1, n}]
(* David Radcliffe, Jun 12 2025 *)

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(k), n//k)
for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 10 2017

Scheme

(define (A286237 n) (A286237bi (A002260 n) (A004736 n)))
(define (A286237bi row col) (if (not (zero? (modulo (+ row col -1) row))) 0 (let ((a (A000010 row)) (b (quotient (+ row col -1) row))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Alternatively, with triangular indexing:
(define (A286237 n) (A286237tr (A002024 n) (A002260 n)))
(define (A286237tr n k) (if (not (zero? (modulo n k))) 0 (let ((a (A000010 k)) (b (/ n k))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Note that: (A286237tr n k) is equal to (A286237bi k (+ 1 (- n k))).

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 + ((A000010(k)+(n/k))^2) - A000010(k) - 3*(n/k)).

T(n,k) = A051731(n,k) * A286235(n,k).

Other identities. For all n >= 1:

T(prime(n),prime(n)) = A000217(prime(n)-1).