A380600 Irregular table T(n, k), n > 0, k = 1..A000005(n) read by rows: the n-th row lists the numbers of the form n * (d-1) / d with d a positive divisor of n.

0, 0, 1, 0, 2, 0, 2, 3, 0, 4, 0, 3, 4, 5, 0, 6, 0, 4, 6, 7, 0, 6, 8, 0, 5, 8, 9, 0, 10, 0, 6, 8, 9, 10, 11, 0, 12, 0, 7, 12, 13, 0, 10, 12, 14, 0, 8, 12, 14, 15, 0, 16, 0, 9, 12, 15, 16, 17, 0, 18, 0, 10, 15, 16, 18, 19, 0, 14, 18, 20, 0, 11, 20, 21, 0, 22

Offset: 1

Author

Rémy Sigrist, Feb 02 2025

Keywords

• nonn
• tabf
• easy

Examples

			Table T(n, k) begins:
  n   n-th row
  --  ------------------
   1  0
   2  0, 1
   3  0, 2
   4  0, 2, 3
   5  0, 4
   6  0, 3, 4, 5
   7  0, 6
   8  0, 4, 6, 7
   9  0, 6, 8
  10  0, 5, 8, 9
  11  0, 10
  12  0, 6, 8, 9, 10, 11
  13  0, 12
  14  0, 7, 12, 13
		

Links

• Index entries for sequences related to divisors

Crossrefs

Cf. A000005, A027750, A046666, A060681, A072513, A094471 (row sums), A258324.

Programs

Mathematica

Table[Map[n*(# - 1)/# &, Divisors[n]], {n, 23}] // Flatten (* Michael De Vlieger, Feb 03 2025 *)

PARI

row(n) = apply (d -> n*(d-1)/d, divisors(n))

Formula

T(n, k) = n * (A027750(n, k) - 1) / A027750(n, k).

Sum_{k = 1..A000005(n)} T(n, k) = A094471(n).

Product_{k = 2..A000005(n)} T(n, k) = A072513(n).

LCM{k = 2..A000005(n)} T(n, k) = A258324(n).

T(n, 1) = 0.

T(n, 2) = A060681(n) for any n > 1. - Michel Marcus, Feb 03 2025

T(n, A000005(n)-1) = A046666(n) for any n > 1.

T(n, A000005(n)) = n-1.