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

1,6

Links

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

Formula

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

Example

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).

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}]

Crossrefs

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

Sequence in context: A351093 A070104 A131085 * A318717 A331887 A328220

Adjacent sequences: A303277 A303278 A303279 * A303281 A303282 A303283

Keyword

nonn

Author

Gus Wiseman, Apr 20 2018

Status

approved