A352549 Irregular table, read by rows: row n lists all numbers equal to n times the number of their divisors.

1, 2, 8, 12, 9, 18, 24, 36, 40, 60, 72, 56, 84, 80, 96, 108, 180, 88, 132, 240, 104, 156, 252, 360, 128, 288, 136, 204, 152, 228, 480, 504, 396, 184, 276, 384, 720, 225, 450, 600, 468, 560, 672, 232, 348, 248, 372, 448, 792, 612, 1260, 864, 296, 444, 684

Offset: 1

Author

M. F. Hasler, Apr 16 2022

Keywords

• nonn
• tabf

Comments

Refactorable or tau-numbers A033950 are those numbers j such that d(j) | j, where d = A000005 is the number of divisors. For any given n = j / d(j), there are only a finite number of solutions to this equation (cf. examples), which are listed in row n of this table.

Examples

			The table starts:
  row n | numbers j such that j = n*A000005(j)
    1   |   1,  2
    2   |   8, 12
    3   |   9, 18, 24
    4   |  36
    5   |  40, 60
    6   |  72
    7   |  56, 84
   ...
If j = p1^e1 * p2^e2 * ... * pK^eK, let d = A000005(j) = (e1+1)*...*(eK+1) for the number of divisors of j (or d(m) for the number of divisors of m).
j = 1 with d = 1 and j = 2 with d = 2 are the only numbers with j/d = 1, listed in row 1.
j = 8 = 2^3 with d = 4 and j = 12 = 2^2*3 with d = 3*2 = 6 are the only numbers with j/d = 2, listed in row 2. Indeed, let j = 2^k*m with odd m, then d = (k+1)*d(m), and j/d = 2 <=> 2^(k-1)*m = (k+1)*d(m), k >= 1. For k = 1, m = 2*d(m), no solution with odd m. For k = 2, 2*m = 3*d(m), the only solution is m = 3, d(m) = 2, j = 12. For k = 3, 4*m = 4*d(m), m = 2 is the only solution. For k > 3, there is no solution: (k+1) will be smaller than 2^(k-1), and for d(m) to have enough powers of 2, m must have 3 (or larger primes) raised to odd powers, but one easily sees that then the l.h.s. is always larger than the r.h.s.
j = 9 = 3^2 with d = 3, j = 18 = 2*3^2 with d = 2*3 = 6, and j = 24 = 2^3*3 with d = 4*2 = 8 are the only numbers with j/d = 3, listed in row 3.
j = 36 = 2^2*3^2 with d = 3*3 is the only number with j/d = 4, listed in row 4.
18 = A036763(1) is the smallest positive integer not of the form j/d(j) for any n, therefore row 18 is empty.
		

Crossrefs

Cf. A000005 (number of divisors), A051521 (row lengths: # {k | k/d(k) = n}), A036763 (indices of empty rows).

Cf. A036764 (first number of row n, or 0 if empty).

Programs

PARI

vecsort(A033950_upto(1300), n->n/numdiv(n))[1..55]