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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. 17

%I

%S 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,

%T 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,

%U 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

%N 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.

%H Alois P. Heinz, <a href="/A077592/b077592.txt">Antidiagonals n = 1..141, flattened</a>

%F 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).

%F Columns are multiplicative.

%F Dirichlet g.f. for column k: Zeta(s)^k. - _Geoffrey Critzer_, Feb 16 2015

%e Rows start:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, 6, 7, ...

%e 1, 2, 3, 4, 5, 6, 7, ...

%e 1, 3, 6, 10, 15, 21, 28, ...

%e 1, 2, 3, 4, 5, 6, 7, ...

%e 1, 4, 9, 16, 25, 36, 49, ...

%e ...

%e 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

%p with(numtheory):

%p A:= proc(n,k) option remember; `if`(k=1, 1,

%p add(A(d, k-1), d=divisors(n)))

%p end:

%p seq(seq(A(n, 1+d-n), n=1..d), d=1..14); # _Alois P. Heinz_, Feb 25 2015

%t 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_ *)

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

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

%Y Main diagonal gives A163767.

%Y Cf. A077593.

%K mult,nonn,tabl,look

%O 1,5

%A _Henry Bottomley_, Nov 08 2002

%E Typo in formula fixed by _Geoffrey Critzer_, Feb 16 2015

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Last modified March 25 15:02 EDT 2019. Contains 321470 sequences. (Running on oeis4.)