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A352549
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Irregular table, read by rows: row n lists all numbers equal to n times the number of their divisors.
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0
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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(PARI) vecsort(A033950_upto(1300), n->n/numdiv(n))[1..55]
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CROSSREFS
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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).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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