A077592 Table by antidiagonals of tau_k(n), the k-th Piltz function (see A007425), or n-th term of the sequence resulting from applying the inverse Möbius transform (k-1) times to the all-ones sequence.

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 6, 2, 1, 1, 6, 5, 10, 3, 4, 1, 1, 7, 6, 15, 4, 9, 2, 1, 1, 8, 7, 21, 5, 16, 3, 4, 1, 1, 9, 8, 28, 6, 25, 4, 10, 3, 1, 1, 10, 9, 36, 7, 36, 5, 20, 6, 4, 1, 1, 11, 10, 45, 8, 49, 6, 35, 10, 9, 2, 1, 1, 12, 11, 55, 9, 64, 7, 56, 15, 16, 3, 6, 1

Offset

1,5

Comments

As an array with offset n=0, k=1, also the number of length n chains of divisors of k. - Gus Wiseman, Aug 04 2022

Links

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.

Wikipedia, Adolf Piltz.

Formula

If n = Product_i p_i^e_i, then T(n,k) = Product_i C(k+e_i-1, e_i). T(n,k) = sum_d{d|n} T(n-1,d) = A077593(n,k) - A077593(n-1,k).

Columns are multiplicative.

Dirichlet g.f. for column k: Zeta(s)^k. - Geoffrey Critzer, Feb 16 2015

A(n,k) = A334997(k,n). - Gus Wiseman, Aug 04 2022

Example

T(6,3) = 9 because we have: 1*1*6, 1*2*3, 1*3*2, 1*6*1, 2*1*3, 2*3*1, 3*1*2, 3*2*1, 6*1*1. - Geoffrey Critzer, Feb 16 2015
From Gus Wiseman, May 03 2021: (Start)
Array begins:
       k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=0:  1   1   1   1   1   1   1   1
  n=1:  1   2   2   3   2   4   2   4
  n=2:  1   3   3   6   3   9   3  10
  n=3:  1   4   4  10   4  16   4  20
  n=4:  1   5   5  15   5  25   5  35
  n=5:  1   6   6  21   6  36   6  56
  n=6:  1   7   7  28   7  49   7  84
  n=7:  1   8   8  36   8  64   8 120
  n=8:  1   9   9  45   9  81   9 165
The triangular form T(n,k) = A(n-k,k) gives the number of length n - k chains of divisors of k. It begins:
  1
  1  1
  1  2  1
  1  3  2  1
  1  4  3  3  1
  1  5  4  6  2  1
  1  6  5 10  3  4  1
  1  7  6 15  4  9  2  1
  1  8  7 21  5 16  3  4  1
  1  9  8 28  6 25  4 10  3  1
  1 10  9 36  7 36  5 20  6  4  1
  1 11 10 45  8 49  6 35 10  9  2  1
(End)

Maple

with(numtheory):
A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=divisors(n)))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # Alois P. Heinz, Feb 25 2015

Mathematica

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[tau[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten (* Robert G. Wilson v *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#] + k - 1, k - 1] & /@ FactorInteger[n]); Table[tau[k, n - k + 1], {n, 1, 13}, {k, 1, n}] // Flatten (* Amiram Eldar, Sep 13 2020 *)
Table[Length[Select[Tuples[Divisors[k], n-k], And@@Divisible@@@Partition[#, 2, 1]&]], {n, 12}, {k, 1, n}] (* TRIANGLE, Gus Wiseman, May 03 2021 *)
Table[Length[Select[Tuples[Divisors[k], n-1], And@@Divisible@@@Partition[#, 2, 1]&]], {n, 6}, {k, 6}] (* ARRAY, Gus Wiseman, May 03 2021 *)

Crossrefs

Rows include: A000012, A000005, A034695, A111217, A111218, A111219, A111220, A111221, A111306.

Columns include (with multiplicity and some offsets) A000012, A000027, A000027, A000217, A000027, A000290, A000027, A000292, A000217, A000290, A000027, A002411, A000027, A000290, A000290, A000332 etc.

Cf. A077593.

Row n = 2 of the array is A007425.

Row n = 3 of the array is A007426.

Row n = 4 of the array is A061200.

The diagonal n = k of the array (central column of the triangle) is A163767.

The transpose of the array is A334997.

Diagonal n = k of the array is A343939.

Antidiagonal sums of the array (or row sums of the triangle) are A343940.

A067824(n) counts strict chains of divisors starting with n.

A074206(n) counts strict chains of divisors from n to 1.

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

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

A253249(n) counts nonempty chains of divisors of n.

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

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

Cf. A018892, A051026, A062319, A143773, A176029, A327527, A337256, A343656, A343658, A343662.

Sequence in context: A243714 A343656 A278427 * A343658 A194005 A055794

Adjacent sequences: A077589 A077590 A077591 * A077593 A077594 A077595

Keyword

mult,nonn,tabl,look

Author

Henry Bottomley, Nov 08 2002

Extensions

Typo in formula fixed by Geoffrey Critzer, Feb 16 2015

Status

approved