A343662 Irregular triangle read by rows where T(n,k) is the number of strict length k chains of divisors of n, 0 <= k <= Omega(n) + 1.

1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 4, 5, 2, 1, 2, 1, 1, 4, 6, 4, 1, 1, 3, 3, 1, 1, 4, 5, 2, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 4, 5, 2, 1, 4, 5, 2, 1, 5, 10, 10, 5, 1, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 6, 12, 10, 3, 1, 4, 5, 2, 1, 4, 5, 2

Offset: 1

Author

Gus Wiseman, May 01 2021

Keywords

• nonn
• tabf

Examples

			Triangle begins:
   1:  1  1
   2:  1  2  1
   3:  1  2  1
   4:  1  3  3  1
   5:  1  2  1
   6:  1  4  5  2
   7:  1  2  1
   8:  1  4  6  4  1
   9:  1  3  3  1
  10:  1  4  5  2
  11:  1  2  1
  12:  1  6 12 10  3
  13:  1  2  1
  14:  1  4  5  2
  15:  1  4  5  2
  16:  1  5 10 10  5  1
For example, row n = 12 counts the following chains:
  ()  (1)   (2/1)   (4/2/1)   (12/4/2/1)
      (2)   (3/1)   (6/2/1)   (12/6/2/1)
      (3)   (4/1)   (6/3/1)   (12/6/3/1)
      (4)   (4/2)   (12/2/1)
      (6)   (6/1)   (12/3/1)
      (12)  (6/2)   (12/4/1)
            (6/3)   (12/4/2)
            (12/1)  (12/6/1)
            (12/2)  (12/6/2)
            (12/3)  (12/6/3)
            (12/4)
            (12/6)
		

Crossrefs

Column k = 1 is A000005.

Row ends are A008480.

Row lengths are A073093.

Column k = 2 is A238952.

The case from n to 1 is A334996 or A251683 (row sums: A074206).

A non-strict version is A334997 (transpose: A077592).

The case starting with n is A337255 (row sums: A067824).

Row sums are A337256 (nonempty: A253249).

A001055 counts factorizations.

A001221 counts distinct prime factors.

A001222 counts prime factors with multiplicity.

A097805 counts compositions by sum and length.

A122651 counts strict chains of divisors summing to n.

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

A163767 counts length n - 1 chains of divisors of n.

A167865 counts strict chains of divisors > 1 summing to n.

A337070 counts strict chains of divisors starting with superprimorials.

Cf. A002033, A007425, A007426, A051026, A062319, A143773, A186972, A327527, A337074, A337105, A337107, A343658.

Programs

Mathematica

Table[Length[Select[Reverse/@Subsets[Divisors[n],{k}],And@@Divisible@@@Partition[#,2,1]&]],{n,15},{k,0,PrimeOmega[n]+1}]