A126131 a(n) = number of divisors of n which equal any d(k) for 1 <= k <= n, where d(k) is the number of positive divisors of k.

1, 2, 1, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 3, 1, 5, 1, 4, 2, 2, 2, 6, 1, 2, 2, 5, 1, 4, 1, 3, 4, 2, 1, 6, 1, 4, 2, 3, 1, 5, 2, 4, 2, 2, 1, 8, 1, 2, 3, 4, 2, 4, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 4, 1, 6, 3, 2, 1, 7, 2, 2, 2, 4, 1, 7, 2, 3, 2, 2, 2, 7, 1, 3, 3, 5, 1, 4, 1, 4, 4

Offset: 1

Author

Leroy Quet, Dec 18 2006

Keywords

• nonn

Examples

			The number of divisors of the integers 1 through 10 form the sequence 1,2,2,3,2,4,2,4,3,4. The divisors of 10 are 1,2,5,10. The divisors of 10 which occur in the sequence of d(k)'s, 1 <= k <= 10, are 1 and 2. So a(10) = 2.
From _Michael De Vlieger_, Oct 10 2017: (Start)
Records and their indices in a(n).
i = index in table
n = index of record r in this sequence
k = index of n in A002182.
MN(n) = rev(A054841(n)) = concatenation of multiplicities of
        prime divisors of n, e.g., MN(60) = "211".
r = record in this sequence.
.
   i       n    k   MN(n)   r
  ----------------------------
   1       1    1   0       1
   2       2    2   1       2
   3       6    4   11      3
   4      12    5   21      5
   5      24    6   31      6
   6      60    9   211     8
   7     120   10   311     9
   8     180   11   221    11
   9     240   12   411    12
  10     360   13   321    14
  11     720   14   421    16
  12    1260   16   2211   18
  13    1680   17   4111   19
  14    2520   18   3211   21
  15    3360        5111   22
  16    5040   19   4211   26
  17    7560   20   3311   28
  18   10080   21   5211   30
  19   15120   22   4311   33
  20   20160   23   6211   34
  21   25200   24   4221   35
  22   30240        5311   38
  23   50400   27   5221   40
  24   60480        6311   42
  25   75600        4321   43
  (End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A126132.

Programs

Mathematica

f[n_] :=Length@ Select[Divisors[n], MemberQ[Table[Length@ Divisors[k], {k, n}], # ] &];Table[f[n], {n, 105}] (* Ray Chandler, Dec 20 2006 *)
Block[{nn = 105, s}, s = DivisorSigma[0, Range@ nn]; Table[DivisorSum[n, 1 &, MemberQ[Take[s, n], #] &], {n, nn}]] (* Michael De Vlieger, Oct 10 2017 *)

PARI

a(n) = #setintersect(divisors(n), Set(vector(n, k, numdiv(k)))); \\ Michel Marcus, Oct 11 2017

Extensions

Extended by Ray Chandler, Dec 20 2006