A077592 -id:A077592 - cp's OEIS frontend

A343940 Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.

1, 2, 4, 7, 12, 19, 30, 45, 66, 95, 135, 187, 256, 346, 463, 613, 803, 1040, 1336, 1703, 2158, 2720, 3409, 4244, 5251, 6461, 7911, 9643, 11707, 14157, 17058, 20480, 24502, 29212, 34707, 41094, 48496, 57053, 66926, 78296, 91369, 106376, 123581, 143276, 165786

Offset: 1

Gus Wiseman, May 07 2021

nonn

			The a(8) = 45 chains:
  ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)  (1/1/1/1/1/1)
      (7)  (2/1)  (5/1/1)  (2/1/1/1)  (3/1/1/1/1)  (2/1/1/1/1/1)
           (2/2)  (5/5/1)  (2/2/1/1)  (3/3/1/1/1)  (2/2/1/1/1/1)
           (3/1)  (5/5/5)  (2/2/2/1)  (3/3/3/1/1)  (2/2/2/1/1/1)
           (3/3)           (2/2/2/2)  (3/3/3/3/1)  (2/2/2/2/1/1)
           (6/1)           (4/1/1/1)  (3/3/3/3/3)  (2/2/2/2/2/1)
           (6/2)           (4/2/1/1)               (2/2/2/2/2/2)
           (6/3)           (4/2/2/1)
           (6/6)           (4/2/2/2)
                           (4/4/1/1)
                           (4/4/2/1)           (1/1/1/1/1/1/1)
                           (4/4/2/2)
                           (4/4/4/1)
                           (4/4/4/2)
                           (4/4/4/4)

Antidiagonal sums of the array (or row sums of the triangle) A334997.
A000005 counts divisors of n.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1.
A146291 counts divisors of n with k prime factors (with multiplicity).
A251683 counts strict length k + 1 chains of divisors from n to 1.
A253249 counts nonempty chains of divisors of n.
A334996 counts strict length k chains of divisors from n to 1.
A337255 counts strict length k chains of divisors starting with n.
Array version of A334997 has:
- column k = 2 A007425,
- transpose A077592,
- subdiagonal n = k + 1 A163767,
- strict case A343662 (row sums: A337256),
- version counting all multisets of divisors (not just chains) A343658,
- diagonal n = k A343939.
Cf. A018892, A062319, A066959, A143773, A176029, A327527, A343656.

Total/@Table[Length[Select[Tuples[Divisors[n-k],k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,0,n-1}]

A306656 Number of ways to fill a 3D matrix with n distinct values.

Original entry on oeis.org

1, 1, 6, 18, 144, 360, 6480, 15120, 403200, 2177280, 32659200, 119750400, 8622028800, 18681062400, 784604620800, 11769069312000, 313841848320000, 1067062284288000, 115242726703104000, 364935301226496000, 43792236147179520000, 459818479545384960000

Offset: 0

Yigit Oktar, Mar 03 2019

nonn

This sequence is a generalization of A323295 to the 3D case. Usually, in multidimensional data related applications (i.e., images, MRI), data is vectorized and then processed. However, because of vectorization, the spatial information in the data is lost. This reverse mapping shows the possible number of spatial states the original data could have been in.

			For n = 6, a(6) = 6480, A007425(6) = 9 namely there are 9 ways to arrange 6 voxels into a 3D matrix: [1,1,6], [1,6,1], [6,1,1], [2,3,1], [3,2,1], [2,1,3], [3,1,2], [1,2,3], [1,3,2]. Then there are 6! ways to fill it with the numbers. 9*6! = 6480.

Cf. A007425, A077592, A323295.

with(numtheory):
a:= n-> `if`(n=0, 1, add(tau(d), d=divisors(n))*n!):
seq(a(n), n=0..23);  # Alois P. Heinz, Mar 03 2019

A007425[n_] := DivisorSigma[0, #]& /@ Divisors[n] // Total;
a[n_] := If[n == 0, 1, A007425[n]*n!];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Feb 02 2025 *)

a(n) = A007425(n) * n! for n > 0, a(0) = 1.

More terms from Alois P. Heinz, Mar 03 2019

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A343940 Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.

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A306656 Number of ways to fill a 3D matrix with n distinct values.

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