A352549 - OEIS

%I #12 Dec 13 2023 03:50:32

%S 1,2,8,12,9,18,24,36,40,60,72,56,84,80,96,108,180,88,132,240,104,156,

%T 252,360,128,288,136,204,152,228,480,504,396,184,276,384,720,225,450,

%U 600,468,560,672,232,348,248,372,448,792,612,1260,864,296,444,684

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

%C 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.

%e The table starts:

%e   row n | numbers j such that j = n*A000005(j)

%e     1   |   1,  2

%e     2   |   8, 12

%e     3   |   9, 18, 24

%e     4   |  36

%e     5   |  40, 60

%e     6   |  72

%e     7   |  56, 84

%e    ...

%e 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).

%e j = 1 with d = 1 and j = 2 with d = 2 are the only numbers with j/d = 1, listed in row 1.

%e 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.

%e 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.

%e j = 36 = 2^2*3^2 with d = 3*3 is the only number with j/d = 4, listed in row 4.

%e 18 = A036763(1) is the smallest positive integer not of the form j/d(j) for any n, therefore row 18 is empty.

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

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

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

%K nonn,tabf

%O 1,2

%A _M. F. Hasler_, Apr 16 2022