A286158 -id:A286158 - cp's OEIS frontend

A286156 A(n,k) = T(remainder(n,k), quotient(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, square array read by descending antidiagonals.

1, 2, 3, 2, 1, 6, 2, 5, 4, 10, 2, 5, 1, 3, 15, 2, 5, 9, 4, 7, 21, 2, 5, 9, 1, 8, 6, 28, 2, 5, 9, 14, 4, 3, 11, 36, 2, 5, 9, 14, 1, 8, 7, 10, 45, 2, 5, 9, 14, 20, 4, 13, 12, 16, 55, 2, 5, 9, 14, 20, 1, 8, 3, 6, 15, 66, 2, 5, 9, 14, 20, 27, 4, 13, 7, 11, 22, 78, 2, 5, 9, 14, 20, 27, 1, 8, 19, 12, 17, 21, 91, 2, 5, 9, 14, 20, 27, 35, 4, 13, 3, 18, 10, 29, 105

Offset: 1

Antti Karttunen, May 04 2017

			The top left 15 X 15 corner of the array:
    1,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,   2,   2
    3,  1,  5,  5,  5,  5,  5,  5,  5,  5,  5,  5,  5,   5,   5
    6,  4,  1,  9,  9,  9,  9,  9,  9,  9,  9,  9,  9,   9,   9
   10,  3,  4,  1, 14, 14, 14, 14, 14, 14, 14, 14, 14,  14,  14
   15,  7,  8,  4,  1, 20, 20, 20, 20, 20, 20, 20, 20,  20,  20
   21,  6,  3,  8,  4,  1, 27, 27, 27, 27, 27, 27, 27,  27,  27
   28, 11,  7, 13,  8,  4,  1, 35, 35, 35, 35, 35, 35,  35,  35
   36, 10, 12,  3, 13,  8,  4,  1, 44, 44, 44, 44, 44,  44,  44
   45, 16,  6,  7, 19, 13,  8,  4,  1, 54, 54, 54, 54,  54,  54
   55, 15, 11, 12,  3, 19, 13,  8,  4,  1, 65, 65, 65,  65,  65
   66, 22, 17, 18,  7, 26, 19, 13,  8,  4,  1, 77, 77,  77,  77
   78, 21, 10,  6, 12,  3, 26, 19, 13,  8,  4,  1, 90,  90,  90
   91, 29, 16, 11, 18,  7, 34, 26, 19, 13,  8,  4,  1, 104, 104
  105, 28, 23, 17, 25, 12,  3, 34, 26, 19, 13,  8,  4,   1, 119
  120, 37, 15, 24,  6, 18,  7, 43, 34, 26, 19, 13,  8,   4,   1

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

Cf. A286157 (transpose),  A286158 (lower triangular region), A286159 (lower triangular region transposed).
Cf. A000217 (column 1), A000012 (the main diagonal), A000096 (superdiagonal), A034856.
Cf. A001477, A285722, A286101, A286102.

Map[((#1 + #2)^2 + 3 #1 + #2)/2 & @@ # & /@ Reverse@ # &, Table[Function[m, Reverse@ QuotientRemainder[m, k]][n - k + 1], {n, 14}, {k, n}]] // Flatten (* Michael De Vlieger, May 20 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n%k, n//k)
for n in range(1, 21): print([A(k, n - k + 1) for k in range(1, n + 1)])  # Indranil Ghosh, May 20 2017

(define (A286156 n) (A286156bi (A002260 n) (A004736 n)))
(define (A286156bi row col) (if (zero? col) -1 (let ((a (remainder row col)) (b (quotient row col))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))))

A(n,k) = T(remainder(n,k), quotient(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...]. This sequence lists only values for indices n >= 1, k >= 1.

A286563 Triangular table T(n,k) read by rows: T(n,1) = 1, and for 1 < k <= n, T(n,k) = the highest exponent e such that k^e divides n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 3, 0, 1, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, May 20 2017

T(n,k) > 0 for k in row n of A027750. - Michael De Vlieger, May 20 2017

Compare rows to those of triangle A279907, smallest exponent e of n divisible by k. The values of k > -1 in row n of A279907 pertain to k in row n of A162306 rather than k in row n of A027750. - Michael De Vlieger, May 21 2017

			The first fifteen rows of this triangular table:
  1,
  1, 1,
  1, 0, 1,
  1, 2, 0, 1,
  1, 0, 0, 0, 1,
  1, 1, 1, 0, 0, 1,
  1, 0, 0, 0, 0, 0, 1,
  1, 3, 0, 1, 0, 0, 0, 1,
  1, 0, 2, 0, 0, 0, 0, 0, 1,
  1, 1, 0, 0, 1, 0, 0, 0, 0, 1,
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1,
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1,
  1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

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

Lower triangular region of A286561.
Cf. A286564 (same triangle reversed).
Cf. A169594 (row sums).
Cf. also arrays A051731, A286158, A027750, A279907, A280269.

T := (n, k) -> ifelse(k = 1, 1, padic:-ordp(n, k)):
for n from 1 to 12 do seq(T(n, k), k = 1..n) od;  # Peter Luschny, Apr 07 2025

Table[If[k == 1, 1, IntegerExponent[n, k]], {n, 15}, {k, n}] // Flatten (* Michael De Vlieger, May 20 2017 *)

def T(n, k):
    i=1
    if k==1: return 1
    while n%(k**i)==0:
        i+=1
    return i-1
for n in range(1, 21): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 20 2017

(define (A286563 n) (A286561bi (A002024 n) (A002260 n))) ;; For A286561bi see A286561.

T(n,k) = A286561(n,k) listed row by row for n >= 1, k = 1 .. n.

cp's OEIS Frontend

A286156 A(n,k) = T(remainder(n,k), quotient(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, square array read by descending antidiagonals.

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