A343940 Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.

1, 2, 4, 7, 12, 19, 30, 45, 66, 95, 135, 187, 256, 346, 463, 613, 803, 1040, 1336, 1703, 2158, 2720, 3409, 4244, 5251, 6461, 7911, 9643, 11707, 14157, 17058, 20480, 24502, 29212, 34707, 41094, 48496, 57053, 66926, 78296, 91369, 106376, 123581, 143276, 165786

Offset

1,2

Links

Table of n, a(n) for n=1..45.

Example

The a(8) = 45 chains:
  ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)  (1/1/1/1/1/1)
      (7)  (2/1)  (5/1/1)  (2/1/1/1)  (3/1/1/1/1)  (2/1/1/1/1/1)
           (2/2)  (5/5/1)  (2/2/1/1)  (3/3/1/1/1)  (2/2/1/1/1/1)
           (3/1)  (5/5/5)  (2/2/2/1)  (3/3/3/1/1)  (2/2/2/1/1/1)
           (3/3)           (2/2/2/2)  (3/3/3/3/1)  (2/2/2/2/1/1)
           (6/1)           (4/1/1/1)  (3/3/3/3/3)  (2/2/2/2/2/1)
           (6/2)           (4/2/1/1)               (2/2/2/2/2/2)
           (6/3)           (4/2/2/1)
           (6/6)           (4/2/2/2)
                           (4/4/1/1)
                           (4/4/2/1)           (1/1/1/1/1/1/1)
                           (4/4/2/2)
                           (4/4/4/1)
                           (4/4/4/2)
                           (4/4/4/4)

Mathematica

Total/@Table[Length[Select[Tuples[Divisors[n-k], k], And@@Divisible@@@Partition[#, 2, 1]&]], {n, 12}, {k, 0, n-1}]

Crossrefs

Antidiagonal sums of the array (or row sums of the triangle) A334997.

A000005 counts divisors of n.

A067824 counts strict chains of divisors starting with n.

A074206 counts strict chains of divisors from n to 1.

A146291 counts divisors of n with k prime factors (with multiplicity).

A251683 counts strict length k + 1 chains of divisors from n to 1.

A253249 counts nonempty chains of divisors of n.

A334996 counts strict length k chains of divisors from n to 1.

A337255 counts strict length k chains of divisors starting with n.

Array version of A334997 has:

- column k = 2 A007425,

- transpose A077592,

- subdiagonal n = k + 1 A163767,

- strict case A343662 (row sums: A337256),

- version counting all multisets of divisors (not just chains) A343658,

- diagonal n = k A343939.

Cf. A018892, A062319, A066959, A143773, A176029, A327527, A343656.

Sequence in context: A035298 A288342 A347543 * A288343 A298414 A347544

Adjacent sequences: A343937 A343938 A343939 * A343941 A343942 A343943

Keyword

nonn

Author

Gus Wiseman, May 07 2021

Status

approved