A319810 - OEIS

%I #7 Sep 29 2018 01:50:00

%S 1,2,2,3,2,5,2,5,4,6,2,11,2,8,7,11,2,17,2,18,9,15,2,32,5,22,12,34,2,

%T 54,2,49,16,51,10,94,2,77,23,112,2,152,2,148,47,165,2,258,7,247,52,

%U 286,2,400,17,402,78,439,2,657,2,594,131,711,24

%N Number of fully periodic integer partitions of n.

%C An integer partition is fully periodic iff either it is a singleton or it is a periodic partition (meaning its multiplicities have a common divisor > 1) with fully periodic multiplicities.

%e The a(12) = 11 fully periodic integer partitions:

%e   (12)

%e   (6,6)

%e   (4,4,4)

%e   (5,5,1,1)

%e   (4,4,2,2)

%e   (3,3,3,3)

%e   (3,3,3,1,1,1)

%e   (3,3,2,2,1,1)

%e   (2,2,2,2,2,2)

%e   (2,2,2,2,1,1,1,1)

%e   (1,1,1,1,1,1,1,1,1,1,1,1)

%e Periodic partitions missing from this list are:

%e   (4,4,1,1,1,1)

%e   (3,3,1,1,1,1,1,1)

%e   (2,2,2,1,1,1,1,1,1)

%e   (2,2,1,1,1,1,1,1,1,1)

%e The first non-uniform fully periodic partition is (4,4,3,3,2,2,2,2,1,1,1,1).

%e The first periodic integer partition that is not fully periodic is (2,2,1,1,1,1).

%t totperQ[m_]:=Or[Length[m]==1,And[GCD@@Length/@Split[Sort[m]]>1,totperQ[Sort[Length/@Split[Sort[m]]]]]];

%t Table[Length[Select[IntegerPartitions[n],totperQ]],{n,30}]

%Y Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319163, A319164, A319811.

%K nonn

%O 1,2

%A _Gus Wiseman_, Sep 28 2018