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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. 18
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, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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

EXAMPLE

Rows start:

  1, 1, 1,  1,  1,  1,  1, ...

  1, 2, 3,  4,  5,  6,  7, ...

  1, 2, 3,  4,  5,  6,  7, ...

  1, 3, 6, 10, 15, 21, 28, ...

  1, 2, 3,  4,  5,  6,  7, ...

  1, 4, 9, 16, 25, 36, 49, ...

  ...

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

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 *)

CROSSREFS

Columns include: A000012, A000005, A007425, A007426, A061200, A034695, A111217, A111218, A111219, A111220, A111221, A111306.

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

Main diagonal gives A163767.

Cf. A077593.

Sequence in context: A193592 A243714 A278427 * A194005 A055794 A092905

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

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Last modified October 16 13:32 EDT 2019. Contains 328093 sequences. (Running on oeis4.)