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.

%I A343940 #9 Mar 28 2024 21:56:40
%S A343940 1,2,4,7,12,19,30,45,66,95,135,187,256,346,463,613,803,1040,1336,1703,
%T A343940 2158,2720,3409,4244,5251,6461,7911,9643,11707,14157,17058,20480,
%U A343940 24502,29212,34707,41094,48496,57053,66926,78296,91369,106376,123581,143276,165786
%N A343940 Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.
%e A343940 The a(8) = 45 chains:
%e A343940   ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)  (1/1/1/1/1/1)
%e A343940       (7)  (2/1)  (5/1/1)  (2/1/1/1)  (3/1/1/1/1)  (2/1/1/1/1/1)
%e A343940            (2/2)  (5/5/1)  (2/2/1/1)  (3/3/1/1/1)  (2/2/1/1/1/1)
%e A343940            (3/1)  (5/5/5)  (2/2/2/1)  (3/3/3/1/1)  (2/2/2/1/1/1)
%e A343940            (3/3)           (2/2/2/2)  (3/3/3/3/1)  (2/2/2/2/1/1)
%e A343940            (6/1)           (4/1/1/1)  (3/3/3/3/3)  (2/2/2/2/2/1)
%e A343940            (6/2)           (4/2/1/1)               (2/2/2/2/2/2)
%e A343940            (6/3)           (4/2/2/1)
%e A343940            (6/6)           (4/2/2/2)
%e A343940                            (4/4/1/1)
%e A343940                            (4/4/2/1)           (1/1/1/1/1/1/1)
%e A343940                            (4/4/2/2)
%e A343940                            (4/4/4/1)
%e A343940                            (4/4/4/2)
%e A343940                            (4/4/4/4)
%t A343940 Total/@Table[Length[Select[Tuples[Divisors[n-k],k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,0,n-1}]
%Y A343940 Antidiagonal sums of the array (or row sums of the triangle) A334997.
%Y A343940 A000005 counts divisors of n.
%Y A343940 A067824 counts strict chains of divisors starting with n.
%Y A343940 A074206 counts strict chains of divisors from n to 1.
%Y A343940 A146291 counts divisors of n with k prime factors (with multiplicity).
%Y A343940 A251683 counts strict length k + 1 chains of divisors from n to 1.
%Y A343940 A253249 counts nonempty chains of divisors of n.
%Y A343940 A334996 counts strict length k chains of divisors from n to 1.
%Y A343940 A337255 counts strict length k chains of divisors starting with n.
%Y A343940 Array version of A334997 has:
%Y A343940 - column k = 2 A007425,
%Y A343940 - transpose A077592,
%Y A343940 - subdiagonal n = k + 1 A163767,
%Y A343940 - strict case A343662 (row sums: A337256),
%Y A343940 - version counting all multisets of divisors (not just chains) A343658,
%Y A343940 - diagonal n = k A343939.
%Y A343940 Cf. A018892, A062319, A066959, A143773, A176029, A327527, A343656.
%K A343940 nonn
%O A343940 1,2
%A A343940 _Gus Wiseman_, May 07 2021