OFFSET
0,3
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750 (terms 0..400 from Alois P. Heinz)
Eric Schmutz, Partitions whose parts are pairwise relatively prime, Discrete Math. 81(1) (1990), 87-89.
Temba Shonhiwa, Compositions with pairwise relatively prime summands within a restricted setting, Fibonacci Quart. 44(4) (2006), 316-323.
FORMULA
log a(n) ~ (2*Pi/sqrt(6)) sqrt(n/log n). - Eric M. Schmidt, Jul 04 2013
Apparently no formula or recurrence is known. - N. J. A. Sloane, Mar 05 2017
EXAMPLE
a(4) = 4 since all partitions of 4 consist of relatively prime numbers except 2+2.
The a(6) = 7 partitions with pairwise coprime parts: (111111), (21111), (3111), (321), (411), (51), (6). - Gus Wiseman, Apr 14 2018
MAPLE
with(numtheory):
b:= proc(n, i, s) option remember; local f;
if n=0 or i=1 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, {}]; Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Oct 03 2013, translated from Maple, after Alois P. Heinz *)
PROG
(Haskell)
a051424 = length . filter f . partitions where
f [] = True
f (p:ps) = (all (== 1) $ map (gcd p) ps) && f ps
partitions n = ps 1 n where
ps x 0 = [[]]
ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 16 2013
KEYWORD
nonn
AUTHOR
EXTENSIONS
More precise definition from Vladeta Jovovic, Dec 11 2004
STATUS
approved