A303280 Number of strict integer partitions of n whose parts have a common divisor other than 1.

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, 32, 13, 38, 7, 57, 1, 54, 19, 68, 1, 95, 1, 90, 33, 104, 1, 148, 5, 149, 39, 166, 1, 230, 14, 226, 55, 256, 1, 360, 1, 340, 82, 390, 20, 527, 1, 513, 105, 609, 1

Offset: 1

Author

Gus Wiseman, Apr 20 2018

Keywords

• nonn

Examples

			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).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> -add(mobius(d)*b(n/d), d=divisors(n) minus {1}):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 23 2018

Mathematica

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

Formula

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