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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.

%I A077592 #49 Jun 01 2025 17:18:37
%S A077592 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 A077592 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 A077592 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
%N 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.
%C A077592 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
%H A077592 Alois P. Heinz, <a href="/A077592/b077592.txt">Antidiagonals n = 1..141, flattened</a>
%H A077592 Adolf Piltz, <a href="https://gdz.sub.uni-goettingen.de/id/PPN271032898">Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst</a>, 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.
%H A077592 Wikipedia, <a href="https://de.wikipedia.org/wiki/Adolf_Piltz">Adolf Piltz</a>.
%F A077592 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 A077592 Columns are multiplicative.
%F A077592 Dirichlet g.f. for column k: Zeta(s)^k. - _Geoffrey Critzer_, Feb 16 2015
%F A077592 A(n,k) = A334997(k,n). - _Gus Wiseman_, Aug 04 2022
%e A077592 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
%e A077592 From _Gus Wiseman_, May 03 2021: (Start)
%e A077592 Array begins:
%e A077592        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
%e A077592   n=0:  1   1   1   1   1   1   1   1
%e A077592   n=1:  1   2   2   3   2   4   2   4
%e A077592   n=2:  1   3   3   6   3   9   3  10
%e A077592   n=3:  1   4   4  10   4  16   4  20
%e A077592   n=4:  1   5   5  15   5  25   5  35
%e A077592   n=5:  1   6   6  21   6  36   6  56
%e A077592   n=6:  1   7   7  28   7  49   7  84
%e A077592   n=7:  1   8   8  36   8  64   8 120
%e A077592   n=8:  1   9   9  45   9  81   9 165
%e A077592 The triangular form T(n,k) = A(n-k,k) gives the number of length n - k chains of divisors of k. It begins:
%e A077592   1
%e A077592   1  1
%e A077592   1  2  1
%e A077592   1  3  2  1
%e A077592   1  4  3  3  1
%e A077592   1  5  4  6  2  1
%e A077592   1  6  5 10  3  4  1
%e A077592   1  7  6 15  4  9  2  1
%e A077592   1  8  7 21  5 16  3  4  1
%e A077592   1  9  8 28  6 25  4 10  3  1
%e A077592   1 10  9 36  7 36  5 20  6  4  1
%e A077592   1 11 10 45  8 49  6 35 10  9  2  1
%e A077592 (End)
%p A077592 with(numtheory):
%p A077592 A:= proc(n,k) option remember; `if`(k=1, 1,
%p A077592       add(A(d, k-1), d=divisors(n)))
%p A077592     end:
%p A077592 seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # _Alois P. Heinz_, Feb 25 2015
%t A077592 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_ *)
%t A077592 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 *)
%t A077592 Table[Length[Select[Tuples[Divisors[k],n-k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,1,n}] (* TRIANGLE, _Gus Wiseman_, May 03 2021 *)
%t A077592 Table[Length[Select[Tuples[Divisors[k],n-1],And@@Divisible@@@Partition[#,2,1]&]],{n,6},{k,6}] (* ARRAY, _Gus Wiseman_, May 03 2021 *)
%Y A077592 Rows include: A000012, A000005, A034695, A111217, A111218, A111219, A111220, A111221, A111306.
%Y A077592 Columns include (with multiplicity and some offsets) A000012, A000027, A000027, A000217, A000027, A000290, A000027, A000292, A000217, A000290, A000027, A002411, A000027, A000290, A000290, A000332 etc.
%Y A077592 Cf. A077593.
%Y A077592 Row n = 2 of the array is A007425.
%Y A077592 Row n = 3 of the array is A007426.
%Y A077592 Row n = 4 of the array is A061200.
%Y A077592 The diagonal n = k of the array (central column of the triangle) is A163767.
%Y A077592 The transpose of the array is A334997.
%Y A077592 Diagonal n = k of the array is A343939.
%Y A077592 Antidiagonal sums of the array (or row sums of the triangle) are A343940.
%Y A077592 A067824(n) counts strict chains of divisors starting with n.
%Y A077592 A074206(n) counts strict chains of divisors from n to 1.
%Y A077592 A146291(n,k) counts divisors of n with k prime factors (with multiplicity).
%Y A077592 A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
%Y A077592 A253249(n) counts nonempty chains of divisors of n.
%Y A077592 A334996(n,k) counts strict length k chains of divisors from n to 1.
%Y A077592 A337255(n,k) counts strict length k chains of divisors starting with n.
%Y A077592 Cf. A018892, A051026, A062319, A143773, A176029, A327527, A337256, A343656, A343658, A343662.
%K A077592 mult,nonn,tabl,look
%O A077592 1,5
%A A077592 _Henry Bottomley_, Nov 08 2002
%E A077592 Typo in formula fixed by _Geoffrey Critzer_, Feb 16 2015