A343147
The number of partitions of the n-th primorial into distinct parts.
Original entry on oeis.org
1, 1, 4, 296, 884987529, 41144767887910339859917073881177514
Offset: 0
-
b:= proc(n) b(n):= `if`(n=0, 1, b(n-1)*ithprime(n)) end:
g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
a:= n-> g(b(n)):
seq(a(n), n=0..5);
-
$RecursionLimit = 2^13;
b[n_] := b[n] = If[n == 0, 1, b[n - 1]*Prime[n]];
g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j]*Sum[
If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
a[n_] := g[b[n]];
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)
A343119
Number of compositions (ordered partitions) of the n-th primorial into distinct parts.
Original entry on oeis.org
1, 1, 11, 41867, 517934206090276988507, 42635439758725572299058305546953458030363703549127905691758491973278624456679699932948789006991639715987
Offset: 0
-
b:= proc(n) b(n):= `if`(n=0, 1, b(n-1)*ithprime(n)) end:
g:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
`if`(k=0, `if`(n=0, 1, 0), g(n-k, k)+k*g(n-k, k-1)))
end:
a:= n-> add(g(b(n), k), k=0..floor((sqrt(8*b(n)+1)-1)/2)):
seq(a(n), n=0..5);
-
$RecursionLimit = 5000;
b[n_] := If[n == 0, 1, b[n - 1]*Prime[n]];
g[n_, k_] := g[n, k] = If[k < 0 || n < 0, 0,
If[k == 0, If[n == 0, 1, 0], g[n - k, k] + k*g[n - k, k - 1]]];
a[n_] := Sum[g[b[n], k], {k, 0, Floor[(Sqrt[8*b[n] + 1] - 1)/2]}];
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)
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