A036764 -id:A036764 - cp's OEIS frontend

A036763 Numbers k such that k*d(x) = x has no solution for x, where d (A000005) is the number of divisors; sequence gives impossible x/d(x) quotients in order of magnitude.

18, 27, 30, 45, 63, 64, 72, 99, 105, 112, 117, 144, 153, 160, 162, 165, 171, 195, 207, 225, 243, 252, 255, 261, 279, 285, 288, 294, 320, 333, 336, 345, 352, 360, 369, 387, 396, 405, 416, 423, 435, 441, 465, 468, 477, 490, 504, 531, 544, 549, 555, 567, 576

Offset: 1

Labos Elemer

nonn

A special case of a bound on d(n) by Erdős and Suranyi (1960) was used to get a limit: a = x/d(x) > 0.5*sqrt(x) and below 4194304 a computer test shows these values did not occur as x = a*d(x). For larger x this is impossible since if d(x) < sqrt(x), then x/d(x) > sqrt(4194304) = 2048 > the given terms.
A051521(a(n)) = 0. - Reinhard Zumkeller, Dec 28 2011
This sequence contains all numbers of the form k = 9p, p prime (i.e., k = 18, 27, 45, 63, 99, ...). - Jianing Song, Nov 25 2018

			No natural number x exists for which x = 18*d(x), so 18 is a term.

P. Erdős and J. Suranyi, Selected Topics in Number Theory, Tankonyvkiado, Budapest, 1960 (in Hungarian).
P. Erdős and J. Suranyi, Selected Topics in Number Theory, Springer, New York, 2003 (in English).

Donovan Johnson, Table of n, a(n) for n = 1..5000

Cf. A000005, A033950, A036761, A036762, A036764, A051278, A051279, A051280, A051521.

a036763 n = a036763_list !! (n-1)
a036763_list = filter ((== 0) . a051521) [1..]
-- Reinhard Zumkeller, Dec 28 2011

with(numtheory): A036763 := proc(n) local k,p: for k from 1 to 4*n^2 do p:=n*k: if(p=n*tau(p))then return NULL: fi: od: return n: end: seq(A036763(n),n=1..100); # Nathaniel Johnston, May 04 2011

noSolQ[n_] := !AnyTrue[Range[4*n^2], # == DivisorSigma[0, n*#]& ];
Reap[Do[If[noSolQ[n], Print[n]; Sow[n]], {n, 600}]][[2, 1]] (* Jean-François Alcover, Jan 30 2018 *)

Definition corrected by N. J. A. Sloane, May 18 2022 at the suggestion of David James Sycamore.

A051521 Number of ways to represent n as k/d(k), where d(k) = A000005(k) is the number of divisors of k.

Original entry on oeis.org

2, 2, 3, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 1, 1, 1, 2, 2, 3, 1, 0, 2, 2, 0, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 2, 0, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, 2, 1, 1, 2, 1, 2, 1, 0, 0, 1, 1, 2, 2, 1, 1, 2, 0, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1

Offset: 1

David W. Wilson

nonn

From Jianing Song, Nov 25 2018: (Start)
a(9p) = 0 for all primes p. Here is a brief proof: a(18) = a(27) = a(45) = a(63) = 0. Now let p be a prime >= 11.
If there is an x such that d(9p*x) = x, let x = p^a*3^b*y, gcd(p, y) = gcd(3, y) = 1, then p^a*3^b*y = d(p^(a+1)*3^(b+2)*y) = (a + 2)*(b + 3)*d(y). Since y >= d(y), we must have (a + 2)*(b + 3) >= p^a*3^b >= 11^a*3^b. If a >= 1, then 3 >= (b + 3)/3^b >= 11^a/(a + 2) >= 11/3, a contradiction. So a = 0. 3^b/(b + 3) <= 2, so b = 0, 1, 2.
Case (i): b = 0, then y = 6*d(y), which has a unique solution y = 72. But gcd(3, 72) != 1, a contradiction,
Case (ii): b = 1, then y = (8/3)*d(y), which has no solution.
Case (iii): b = 2, then y = (10/9)*d(y), which has no solution.
Similarly, it can be proved that a(81p) = 0 for all primes p. (End)

			There are a(1) = 2 numbers k for which k/d(k) = 1, namely k = 1 and k = 2.
There are a(2) = 2 numbers k for which k/d(k) = 2, namely k = 8 and k = 12.
There are a(3) = 3 numbers k for which k/d(k) = 3, namely k = 9, 18 and 24.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000005 (number of divisors), A033950, A036762, A036763 (indices of 0s), A036764, A051278 (indices of 1s), A051279 (indices of 2s).

a051521 n = length [k | k <- [1..4*n^2],
                        let d = a000005 k, divMod k d == (n,0)]
-- Reinhard Zumkeller, Dec 28 2011

a[n_] := Count[Table[n == k/DivisorSigma[0, k], {k, 1, 4*n^2}], True]; Table[a[n], {n, 1, 100}]  (* Jean-François Alcover, Oct 22 2012 *)

a(A036763(n)) = 0; a(A051278(n)) = 1; a(A051279(n)) = 2. - Reinhard Zumkeller, Dec 28 2011

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

Original entry on oeis.org

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

M. F. Hasler, Apr 16 2022

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.

			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.

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).

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

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A036763 Numbers k such that k*d(x) = x has no solution for x, where d (A000005) is the number of divisors; sequence gives impossible x/d(x) quotients in order of magnitude.

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A051521 Number of ways to represent n as k/d(k), where d(k) = A000005(k) is the number of divisors of k.

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

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