A348551 Heinz numbers of integer partitions whose mean is not an integer.

1, 6, 12, 14, 15, 18, 20, 24, 26, 33, 35, 36, 38, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 58, 60, 63, 65, 66, 69, 70, 72, 74, 75, 76, 77, 80, 86, 92, 93, 95, 96, 102, 104, 106, 108, 112, 114, 117, 119, 120, 122, 123, 124, 126, 130, 132, 135, 136, 140, 141, 142

Offset: 1

Author

Gus Wiseman, Nov 14 2021

Keywords

• nonn

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.

Examples

			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}
		

Crossrefs

A version counting nonempty subsets is A000079 - A051293.

A version counting factorizations is A001055 - A326622.

A version counting compositions is A011782 - A271654.

A version for prime factors is A175352, complement A078175.

A version for distinct prime factors A176587, complement A078174.

The complement is A316413, counted by A067538, strict A102627.

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.

A018818 counts partitions into divisors, ranked by A326841.

A143773 counts partitions into multiples of the length, ranked by A316428.

A236634 counts unbalanced partitions.

A047993 counts balanced partitions, ranked by A106529.

A056239 adds up prime indices, row sums of A112798.

A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.

A327472 counts partitions not containing their mean, complement A237984.

Cf. A067539, A096199, A098743, A175397, A175761, A289508, A289509, A290103, A326028, A326645, A326837.

Programs

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[#]]]&]