A303280 - OEIS

%I #11 Apr 29 2018 18:53:35

%S 0,1,1,1,1,2,1,2,2,3,1,5,1,5,4,6,1,10,1,11,6,12,1,19,3,18,8,23,1,36,1,

%T 32,13,38,7,57,1,54,19,68,1,95,1,90,33,104,1,148,5,149,39,166,1,230,

%U 14,226,55,256,1,360,1,340,82,390,20,527,1,513,105,609,1

%N Number of strict integer partitions of n whose parts have a common divisor other than 1.

%H Alois P. Heinz, <a href="/A303280/b303280.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = -Sum_{d|n, d > 1} mu(d) * A000009(n/d).

%e The a(18) = 10 strict partitions are (18), (10,8), (12,6), (14,4), (15,3), (16,2), (8,6,4), (9,6,3), (10,6,2), (12,4,2).

%p with(numtheory):

%p b:= proc(n) option remember; `if`(n=0, 1, add(add(

%p       `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)

%p     end:

%p a:= n-> -add(mobius(d)*b(n/d), d=divisors(n) minus {1}):

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Apr 23 2018

%t Table[-Sum[MoebiusMu[d]*PartitionsQ[n/d],{d,Rest[Divisors[n]]}],{n,100}]

%Y Cf. A000009, A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302698, A302796, A303138.

%K nonn

%O 1,6

%A _Gus Wiseman_, Apr 20 2018