OFFSET
1,3
COMMENTS
a(p) = p!-(p-1)! for prime p. Conjecture: a(n) is divisible by n^2-1 for n>3.
Conjecture: gcd(a(n),n)=1. - Vladeta Jovovic, Jan 25 2003
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..450
MAPLE
with(combinat):
b:= proc(n, i, g) option remember; `if`(n=0, `if`(g>=2, 1, 0),
`if`(i<2, 0, b(n, i-1, g) +`if`(igcd(g, i)<2, 0,
add((i-1)!^j/j! *multinomial(n, i$j, n-i*j)*
b(n-i*j, i-1, igcd(i, g)), j=1..n/i))))
end:
a:= n-> n!-b(n, n, 0):
seq(a(n), n=1..25); # Alois P. Heinz, Jun 06 2013
# second Maple program:
b:= proc(n, g) option remember; `if`(n=0, `if`(g=1, 1, 0), add(
(j-1)!*b(n-j, igcd(g, j))*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=1..25); # Alois P. Heinz, Jul 03 2021
MATHEMATICA
f[list_] :=
Total[list]!/Apply[Times, Table[list[[i]], {i, 1, Length[list]}]]/
Apply[Times, Select[Table[Count[list, i], {i, 1, Total[list]}], # > 0 &]!];
Table[Total[Map[f, Select[IntegerPartitions[n], Apply[GCD, #] == 1 &]]], {n, 1, 25}] (* Geoffrey Critzer, Jun 06 2013 *)
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, g_] := b[n, i, g] = If[n==0, If[g >= 2, 1, 0], If[i<2, 0, b[n, i-1, g] + If[GCD[g, i]<2, 0, Sum[(i-1)!^j/j!*multinomial[n, Append[Array[i&, j], n-i*j]]*b[n-i*j, i-1, GCD[i, g]], {j, 1, n/i}]]]]; a[n_] := n! - b[n, n, 0]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Vladimir Baltic, Dec 27 2002
STATUS
approved