OFFSET
0,6
COMMENTS
This sequence is of interest for group theory. The partitions counted by a(n) correspond to conjugacy classes of optimal order of the symmetric group of n elements: they have no fixed point, their order is the direct product of their cycle lengths and they are not contained in a subgroup of Sym_p for p < n. A123131 gives the maximum order (LCM) reachable by these partitions.
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750 (terms 0..400 from Alois P. Heinz)
M. LeBrun & D. Hoey, Emails
FORMULA
EXAMPLE
The a(17) = 9 strict partitions into pairwise coprime parts that are greater than 1 are (17), (15,2), (14,3), (13,4), (12,5), (11,6), (10,7), (9,8), (7,5,3,2). - Gus Wiseman, Apr 14 2018
MAPLE
with(numtheory):
b:= proc(n, i, s) option remember; local f;
if n=0 then 1
elif i<2 then 0
else f:= factorset(i);
b(n, i-1, select(x-> is(x<i), s)) +`if`(i<=n and f intersect s={},
b(n-i, i-1, select(x-> is(x<i), s union f)), 0)
fi
end:
a:= n-> b(n, n, {}):
seq(a(n), n=0..80); # Alois P. Heinz, Mar 14 2012
MATHEMATICA
b[n_, i_, s_] := b[n, i, s] = Module[{f}, If[n == 0 || i == 1, 1, If[i<2, 0, f = FactorInteger[i][[All, 1]]; b[n, i-1, Select[s, #<i&]]+If[i <= n && f ~Intersection~ s == {}, b[n-i, i-1, Select[s ~Union~ f, #<i&]], 0]]]]; a[n_] := b[n, n, {}]-b[n-1, n-1, {}]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n], FreeQ[#, 1]&&(Length[#]===1||CoprimeQ@@#)&]], {n, 20}] (* Gus Wiseman, Apr 14 2018 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More precise definition from Vladeta Jovovic, Dec 11 2004
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 13 2005
STATUS
approved