A337255 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors starting with n.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 5, 7, 3, 1, 1, 1, 3, 2, 1, 3, 2, 1, 4, 6, 4, 1, 1, 1, 1, 5, 7, 3, 1, 1, 1, 5, 7, 3, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 15, 13, 4, 1, 2, 1, 1, 3, 2, 1, 3, 3, 1, 1, 5, 7, 3, 1, 1, 1

Offset

1,7

Links

Alois P. Heinz, Rows n = 1..5000, flattened

Example

Sequence of rows begins:
     1: {1}           16: {1,4,6,4,1}
     2: {1,1}         17: {1,1}
     3: {1,1}         18: {1,5,7,3}
     4: {1,2,1}       19: {1,1}
     5: {1,1}         20: {1,5,7,3}
     6: {1,3,2}       21: {1,3,2}
     7: {1,1}         22: {1,3,2}
     8: {1,3,3,1}     23: {1,1}
     9: {1,2,1}       24: {1,7,15,13,4}
    10: {1,3,2}       25: {1,2,1}
    11: {1,1}         26: {1,3,2}
    12: {1,5,7,3}     27: {1,3,3,1}
    13: {1,1}         28: {1,5,7,3}
    14: {1,3,2}       29: {1,1}
    15: {1,3,2}       30: {1,7,12,6}
Row n = 24 counts the following chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/6/2/1   24/12/4/2/1
      24/3   24/4/1   24/6/3/1   24/12/6/2/1
      24/4   24/4/2   24/8/2/1   24/12/6/3/1
      24/6   24/6/1   24/8/4/1
      24/8   24/6/2   24/8/4/2
      24/12  24/6/3   24/12/2/1
             24/8/1   24/12/3/1
             24/8/2   24/12/4/1
             24/8/4   24/12/4/2
             24/12/1  24/12/6/1
             24/12/2  24/12/6/2
             24/12/3  24/12/6/3
             24/12/4
             24/12/6

Maple

b:= proc(n) option remember; expand(x*(1 +
      add(b(d), d=numtheory[divisors](n) minus {n})))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=1..50);  # Alois P. Heinz, Aug 23 2020

Mathematica

chss[n_]:=Prepend[Join@@Table[Prepend[#, n]&/@chss[d], {d, Most[Divisors[n]]}], {n}];
Table[Length[Select[chss[n], Length[#]==k&]], {n, 30}, {k, 1+PrimeOmega[n]}]

Crossrefs

A008480 gives rows ends.

A067824 gives row sums.

A073093 gives row lengths.

A334996 appears to be the case of chains ending with 1.

A337071 is the sum of row n!.

A000005 counts divisors.

A001055 counts factorizations.

A001222 counts prime factors with multiplicity.

A067824 counts chains of divisors starting with n.

A074206 counts chains of divisors from n to 1.

A122651 counts chains of divisors summing to n.

A167865 counts chains of divisors > 1 summing to n.

A251683 counts chains of divisors from n to 1 by length.

A253249 counts nonempty chains of divisors.

A337070 counts chains of divisors starting with A006939(n).

A337256 counts chains of divisors.

Cf. A001221, A002033, A124010, A124433, A337074, A337105, A337107.

Sequence in context: A370770 A124010 A212171 * A337375 A196228 A133364

Adjacent sequences: A337252 A337253 A337254 * A337256 A337257 A337258

Keyword

nonn,tabf

Author

Gus Wiseman, Aug 23 2020

Status

approved