OFFSET
1,2
COMMENTS
Equivalently, partitions whose length does not divide their sum.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The terms and their prime indices begin:
1: {}
6: {1,2}
12: {1,1,2}
14: {1,4}
15: {2,3}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
26: {1,6}
33: {2,5}
35: {3,4}
36: {1,1,2,2}
38: {1,8}
40: {1,1,1,3}
42: {1,2,4}
44: {1,1,5}
45: {2,2,3}
48: {1,1,1,1,2}
MAPLE
q:= n-> (l-> nops(l)=0 or irem(add(i, i=l), nops(l))>0)(map
(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..142])[]; # Alois P. Heinz, Nov 19 2021
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], !IntegerQ[Mean[primeMS[#]]]&]
CROSSREFS
The geometric version is the complement of A326623.
The conjugate version is the complement of A326836.
These partitions are counted by A349156.
A000041 counts partitions.
A001222 counts prime factors with multiplicity.
A236634 counts unbalanced partitions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 14 2021
STATUS
approved