A286234 -id:A286234 - cp's OEIS frontend

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.

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).

A286244 Square array A(n,k) = P(A046523(k), floor((n+k-1)/k)), 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, 3, 2, 3, 3, 4, 10, 3, 5, 7, 3, 10, 3, 5, 11, 21, 3, 10, 5, 8, 16, 3, 21, 3, 10, 5, 8, 22, 36, 3, 21, 3, 14, 5, 12, 29, 10, 36, 3, 21, 3, 14, 8, 12, 37, 21, 10, 36, 3, 21, 5, 14, 8, 17, 46, 3, 21, 10, 36, 3, 21, 5, 14, 8, 17, 56, 78, 3, 21, 10, 36, 3, 27, 5, 19, 12, 23, 67, 3, 78, 3, 21, 10, 36, 3, 27, 5, 19, 12, 23, 79

Offset: 1

Antti Karttunen, May 06 2017

Transpose of A286245.

			The top left 12 X 12 corner of the array:
   1,  3,  3, 10, 3, 21, 3, 36, 10, 21, 3, 78
   2,  3,  3, 10, 3, 21, 3, 36, 10, 21, 3, 78
   4,  5,  3, 10, 3, 21, 3, 36, 10, 21, 3, 78
   7,  5,  5, 10, 3, 21, 3, 36, 10, 21, 3, 78
  11,  8,  5, 14, 3, 21, 3, 36, 10, 21, 3, 78
  16,  8,  5, 14, 5, 21, 3, 36, 10, 21, 3, 78
  22, 12,  8, 14, 5, 27, 3, 36, 10, 21, 3, 78
  29, 12,  8, 14, 5, 27, 5, 36, 10, 21, 3, 78
  37, 17,  8, 19, 5, 27, 5, 44, 10, 21, 3, 78
  46, 17, 12, 19, 5, 27, 5, 44, 14, 21, 3, 78
  56, 23, 12, 19, 8, 27, 5, 44, 14, 27, 3, 78
  67, 23, 12, 19, 8, 27, 5, 44, 14, 27, 5, 78
The first fifteen rows when viewed as a triangle:
   1,
   3,  2,
   3,  3,  4,
  10,  3,  5,  7,
   3, 10,  3,  5, 11,
  21,  3, 10,  5,  8, 16,
   3, 21,  3, 10,  5,  8, 22,
  36,  3, 21,  3, 14,  5, 12, 29,
  10, 36,  3, 21,  3, 14,  8, 12, 37,
  21, 10, 36,  3, 21,  5, 14,  8, 17, 46,
   3, 21, 10, 36,  3, 21,  5, 14,  8, 17, 56,
  78,  3, 21, 10, 36,  3, 27,  5, 19, 12, 23, 67,
   3, 78,  3, 21, 10, 36,  3, 27,  5, 19, 12, 23, 79,
  21,  3, 78,  3, 21, 10, 36,  5, 27,  5, 19, 12, 30, 92,
  21, 21,  3, 78,  3, 21, 10, 36,  5, 27,  8, 19, 17, 30, 106

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

Transpose: A286245.

Cf. A000027, A046523, A286156, A286246, A286234.

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 A(n, k): return T(a046523(k), int((n + k - 1)//k))
for n in range(1, 21): print([A(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, May 09 2017

(define (A286244 n) (A286244bi (A002260 n) (A004736 n)))
(define (A286244bi row col) (let ((a (A046523 col)) (b (quotient (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2))))

cp's OEIS Frontend

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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Mathematica

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Scheme

Formula

A286244 Square array A(n,k) = P(A046523(k), floor((n+k-1)/k)), 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Scheme