A207375 Irregular array read by rows in which row n lists the (one or two) central divisors of n in increasing order.

1, 1, 2, 1, 3, 2, 1, 5, 2, 3, 1, 7, 2, 4, 3, 2, 5, 1, 11, 3, 4, 1, 13, 2, 7, 3, 5, 4, 1, 17, 3, 6, 1, 19, 4, 5, 3, 7, 2, 11, 1, 23, 4, 6, 5, 2, 13, 3, 9, 4, 7, 1, 29, 5, 6, 1, 31, 4, 8, 3, 11, 2, 17, 5, 7, 6, 1, 37, 2, 19, 3, 13, 5, 8, 1, 41, 6, 7, 1, 43

Offset: 1

Author

Omar E. Pol, Feb 23 2012

Keywords

• nonn
• tabf

Comments

If n is a square then row n lists only the square root of n because the squares (A000290) have only one central divisor.

If n is not a square then row n lists the pair (j, k) of divisors of n, nearest to the square root of n, such that j*k = n.

Conjecture 1: It appears that the n-th record in this sequence is the last member of row A008578(n).

Column 1 gives A033676. Right border gives A033677. - Omar E. Pol, Feb 26 2019

The conjecture 1 follows from Bertrand's Postulate. - Charles R Greathouse IV, Feb 11 2022

Row products give A097448. - Omar E. Pol, Feb 17 2022

Examples

			Array begins:
  1;
  1,  2;
  1,  3;
  2;
  1,  5;
  2,  3;
  1,  7;
  2,  4;
  3;
  2,  5;
  1, 11;
  3,  4;
  1, 13;
...
		

Links

• Alois P. Heinz, Rows n = 1..5000
• Omar E. Pol, Illustration of the divisors of the first 12 positive integers

Crossrefs

Row n has length A169695(n).

Row sums give A207376.

Cf. A000005, A000290, A008578, A027750, A033676, A033677, A097448, A161901, A161904.

Programs

Mathematica

A207375row[n_] := ArrayPad[#, -Floor[(Length[#] - 1)/2]] & [Divisors[n]];
Array[A207375row, 50] (* Paolo Xausa, Apr 07 2025 *)