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
Keywords
Examples
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)
Crossrefs
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,
- version counting all multisets of divisors (not just chains) A343658,
- diagonal n = k A343939.
Programs
-
Mathematica
Total/@Table[Length[Select[Tuples[Divisors[n-k],k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,0,n-1}]