A286237 -id:A286237 - cp's OEIS frontend

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

Antti Karttunen, May 06 2017

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.

			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.

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

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).

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

(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)))))

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).

A286235 Triangular table T(n,k) = P(phi(k), floor(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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 05 2017

Equally: square array A(n,k) = P(A000010(n), floor((n+k-1)/n)), 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.

			The first fifteen rows of the triangle:
    1,
    2,  1,
    4,  1,  3,
    7,  2,  3, 3,
   11,  2,  3, 3, 10,
   16,  4,  5, 3, 10, 3,
   22,  4,  5, 3, 10, 3, 21,
   29,  7,  5, 5, 10, 3, 21, 10,
   37,  7,  8, 5, 10, 3, 21, 10, 21,
   46, 11,  8, 5, 14, 3, 21, 10, 21, 10,
   56, 11,  8, 5, 14, 3, 21, 10, 21, 10, 55,
   67, 16, 12, 8, 14, 5, 21, 10, 21, 10, 55, 10,
   79, 16, 12, 8, 14, 5, 21, 10, 21, 10, 55, 10, 78,
   92, 22, 12, 8, 14, 5, 27, 10, 21, 10, 55, 10, 78, 21,
  106, 22, 17, 8, 19, 5, 27, 10, 21, 10, 55, 10, 78, 21, 36

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array
Eric Weisstein's World of Mathematics, Pairing Function

Transpose: A286234.
Cf. A000010, A000027, A286156, A286245.
Cf. A286237 (same triangle but with zeros in positions where k does not divide n).

Map[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ # & /@ # &, Table[{EulerPhi@ k, Floor[n/k]}, {n, 14}, {k, n}]] // Flatten (* Michael De Vlieger, May 06 2017 *)

from sympy import totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def t(n, k): return T(totient(k), int(n//k))
for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 11 2017

(define (A286235 n) (A286235bi (A002260 n) (A004736 n)))
(define (A286235bi row col) (let ((a (A000010 row)) (b (quotient (+ row col -1) row))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2))))

As a triangle (with n >= 1, 1 <= k <= n):

T(n,k) = (1/2)*(2 + ((A000010(k)+floor(n/k))^2) - A000010(k) - 3*floor(n/k)).

A286236 Square array A(n,k) = P(A000010(k), (n+k-1)/k) if k 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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 05 2017

This is transpose of A286237, see comments there.

			The top left 12 X 12 corner of the array:
   1,  1,  3,  3, 10, 3, 21, 10, 21, 10, 55, 10
   2,  0,  0,  0,  0, 0,  0,  0,  0,  0,  0,  0
   4,  2,  0,  0,  0, 0,  0,  0,  0,  0,  0,  0
   7,  0,  5,  0,  0, 0,  0,  0,  0,  0,  0,  0
  11,  4,  0,  5,  0, 0,  0,  0,  0,  0,  0,  0
  16,  0,  0,  0, 14, 0,  0,  0,  0,  0,  0,  0
  22,  7,  8,  0,  0, 5,  0,  0,  0,  0,  0,  0
  29,  0,  0,  0,  0, 0, 27,  0,  0,  0,  0,  0
  37, 11,  0,  8,  0, 0,  0, 14,  0,  0,  0,  0
  46,  0, 12,  0,  0, 0,  0,  0, 27,  0,  0,  0
  56, 16,  0,  0, 19, 0,  0,  0,  0, 14,  0,  0
  67,  0,  0,  0,  0, 0,  0,  0,  0,  0, 65,  0
The first 15 rows when viewed as a triangle:
   1,
   1, 2,
   3, 0, 4,
   3, 0, 2, 7,
  10, 0, 0, 0, 11,
   3, 0, 0, 5,  4, 16,
  21, 0, 0, 0,  0,  0, 22,
  10, 0, 0, 0,  5,  0,  7, 29,
  21, 0, 0, 0,  0,  0,  8,  0, 37,
  10, 0, 0, 0,  0, 14,  0,  0, 11, 46,
  55, 0, 0, 0,  0,  0,  0,  0,  0,  0, 56,
  10, 0, 0, 0,  0,  0,  5,  0,  8, 12, 16, 67,
  78, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0, 79,
  21, 0, 0, 0,  0,  0,  0, 27,  0,  0,  0,  0, 22, 92,
  36, 0, 0, 0,  0,  0,  0,  0,  0,  0, 19,  0, 17,  0, 106

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 rows of triangle/antidiagonals of array

Transpose: A286237.

Cf. A000010, A000027, A113998, A286156, A286234, A286246.

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[Reverse[t[n, #] & /@ Range[n]], {n, 1, 20}] (* David Radcliffe, Jun 12 2025 *)

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)][::-1]) # Indranil Ghosh, May 10 2017

(define (A286236 n) (A286236bi (A002260 n) (A004736 n)))
(define (A286236bi row col) (if (not (zero? (modulo (+ row col -1) col))) 0 (let ((a (A000010 col)) (b (/ (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Alternatively, with triangular indexing:
(define (A286236 n) (A286236tr (A002024 n) (A002260 n)))
(define (A286236tr n k) (A286236bi k (+ 1 (- n k))))

T(n,k) = A113998(n,k) * A286234(n,k).

A286239 Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(A000010(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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 06 2017

This sequence packs the values of phi(n/k) and k (whenever k divides n) 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)
Triangle A286237 has the same property.

			The first fifteen rows of triangle:
   1,
   1,  2,
   3,  0,  4,
   3,  2,  0,  7,
  10,  0,  0,  0, 11,
   3,  5,  4,  0,  0, 16,
  21,  0,  0,  0,  0,  0, 22,
  10,  5,  0,  7,  0,  0,  0, 29,
  21,  0,  8,  0,  0,  0,  0,  0, 37,
  10, 14,  0,  0, 11,  0,  0,  0,  0, 46,
  55,  0,  0,  0,  0,  0,  0,  0,  0,  0, 56,
  10,  5,  8, 12,  0, 16,  0,  0,  0,  0,  0, 67,
  78,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 79,
  21, 27,  0,  0,  0,  0, 22,  0,  0,  0,  0,  0,  0, 92,
  36,  0, 19,  0, 17,  0,  0,  0,  0,  0,  0,  0,  0,  0, 106
   -------------------------------------------------------------
Note how triangle A286237 contains on each row the same numbers in the same "divisibility-allotted" positions, but in reverse order.

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 rows of triangle/antidiagonals of array

Transpose: A286238.
Cf. A000124 (the right edge of the triangle).
Cf. A000010, A000027, A053635, A054610, A054523, A286156, A286237, A286249.

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(n//k), k)
for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 09 2017

(define (A286239 n) (A286239tr (A002024 n) (A002260 n)))
(define (A286239tr n k) (if (not (zero? (modulo n k))) 0 (let ((a (A000010 (/ n k))) (b k)) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))

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

cp's OEIS Frontend

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.

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A286235 Triangular table T(n,k) = P(phi(k), floor(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.

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A286236 Square array A(n,k) = P(A000010(k), (n+k-1)/k) if k 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.

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A286239 Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(A000010(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.

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