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.
Original entry on oeis.org
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
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).
-
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 *)
A051278
Numbers n such that n = k/d(k) has a unique solution, where d(k) = number of divisors of k.
Original entry on oeis.org
4, 6, 9, 10, 12, 14, 15, 20, 21, 22, 26, 32, 33, 34, 35, 36, 38, 39, 42, 46, 50, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 75, 77, 78, 82, 85, 86, 87, 90, 91, 93, 94, 95, 96, 98, 100, 102, 106, 108, 110, 111, 114, 115, 118, 119, 122, 123, 126, 128, 129, 130
Offset: 1
36 is the unique number k with k/d(k)=4.
-
a051278 n = a051278_list !! (n-1)
a051278_list = filter ((== 1) . a051521) [1..]
-- Reinhard Zumkeller, Dec 28 2011
-
with(numtheory): A051278 := proc(n) local ct,k: ct:=0: for k from 1 to 4*n^2 do if(n=k/tau(k))then ct:=ct+1: fi: od: if(ct=1)then return n: else return NULL: fi: end: seq(A051278(n),n=1..40);
-
cnt[n_] := Count[Table[n == k/DivisorSigma[0, k], {k, 1, 4*n^2}], True]; Select[Range[130], cnt[#] == 1 &] (* Jean-François Alcover, Oct 22 2012 *)
A051279
Numbers n such that n = k/d(k) has exactly 2 solutions, where d(k) = number of divisors of k.
Original entry on oeis.org
1, 2, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 28, 29, 31, 37, 41, 43, 44, 47, 48, 52, 53, 56, 59, 61, 67, 68, 71, 73, 76, 79, 80, 81, 83, 84, 88, 89, 92, 97, 101, 103, 104, 107, 109, 113, 116, 120, 124, 127, 131, 132, 136, 137, 139, 148, 149, 151, 152, 154, 156
Offset: 1
There are exactly 2 numbers k, 40 and 60, with k/d(k)=5.
-
a051279 n = a051279_list !! (n-1)
a051279_list = filter ((== 2) . a051521) [1..]
-- Reinhard Zumkeller, Dec 28 2011
-
with(numtheory): A051279 := proc(n) local ct, k: ct:=0: for k from 1 to 4*n^2 do if(n=k/tau(k))then ct:=ct+1: fi: od: if(ct=2)then return n: else return NULL: fi: end: seq(A051279(n), n=1..40); # Nathaniel Johnston, May 04 2011
-
A051279 = Reap[Do[ct = 0; For[k = 1, k <= 4*n^2, k++, If[n == k/DivisorSigma[0, k], ct++]]; If[ct == 2, Print[n]; Sow[n]], {n, 1, 160}]][[2, 1]](* Jean-François Alcover, Apr 16 2012, after Nathaniel Johnston *)
A036764
If n can be expressed as m/d(m) for some m, where d(m) is the number of divisors of m (A000005), then a(n) is the smallest such m, otherwise a(n) = 0.
Original entry on oeis.org
1, 8, 9, 36, 40, 72, 56, 80, 108, 180, 88, 240, 104, 252, 360, 128, 136, 0, 152, 480, 504, 396, 184, 384, 225, 468, 0, 560, 232, 0, 248, 448, 792, 612, 1260, 864, 296, 684, 936, 640, 328, 1680, 344, 880, 0, 828, 376, 1152, 441, 1800, 1224, 1040, 424, 972, 1980
Offset: 1
If q=25 then 25*9 = 225, 25*18 = 450 and 25*24 = 600 so that d(225), d(450), d(600) are 9, 18, 24, respectively. The smallest is 225. Thus a(25)=225.
-
with(numtheory): A036764 := proc(n) local k,p: for k from 1 to 4*n^2 do p:=n*k: if(p=n*tau(p))then return p: fi: od: return 0: end: seq(A036764(n),n=1..40); # Nathaniel Johnston, May 04 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
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).
Showing 1-5 of 5 results.
Comments