OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..7000
FORMULA
a(n) = Sum_y A000005(k), where the sum is over all strict integer partitions of n and k is the number of parts.
EXAMPLE
The a(10) = 20 split partitions:
[10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
.
[9] [8] [7] [6] [4 3]
[1] [2] [3] [4] [2 1]
.
[7] [6] [5] [5]
[2] [3] [4] [3]
[1] [1] [1] [2]
.
[4]
[3]
[2]
[1]
MAPLE
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, numtheory[tau](t), b(n, i-1, t)+
b(n-i, min(n-i, i-1), t+1)))
end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..60); # Alois P. Heinz, Jan 15 2019
MATHEMATICA
Table[Sum[Length[Divisors[Length[ptn]]], {ptn, Select[IntegerPartitions[n], UnsameQ@@#&]}], {n, 30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n>i(i+1)/2, 0,
If[n == 0, DivisorSigma[0, t], b[n, i-1, t] +
b[n-i, Min[n-i, i-1], t+1]]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 60] (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2019
STATUS
approved