A384349 Heinz numbers of integer partitions with no proper way to choose disjoint strict partitions of each part.

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 105, 108, 110, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135

Offset

1,2

Comments

By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.

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.

Links

Table of n, a(n) for n=1..63.

Example

The prime indices of 102 are {1,2,7}, which has proper disjoint choice ((1),(2),(3,4)), so 102 is not in the sequence.
The terms together with their prime indices begin:
     1: {}           27: {2,2,2}        63: {2,2,4}
     2: {1}          28: {1,1,4}        64: {1,1,1,1,1,1}
     3: {2}          30: {1,2,3}        66: {1,2,5}
     4: {1,1}        32: {1,1,1,1,1}    68: {1,1,7}
     6: {1,2}        36: {1,1,2,2}      70: {1,3,4}
     8: {1,1,1}      40: {1,1,1,3}      72: {1,1,1,2,2}
     9: {2,2}        42: {1,2,4}        75: {2,3,3}
    10: {1,3}        44: {1,1,5}        76: {1,1,8}
    12: {1,1,2}      45: {2,2,3}        78: {1,2,6}
    14: {1,4}        48: {1,1,1,1,2}    80: {1,1,1,1,3}
    15: {2,3}        50: {1,3,3}        81: {2,2,2,2}
    16: {1,1,1,1}    52: {1,1,6}        84: {1,1,2,4}
    18: {1,2,2}      54: {1,2,2,2}      88: {1,1,1,5}
    20: {1,1,3}      56: {1,1,1,4}      90: {1,2,2,3}
    24: {1,1,1,2}    60: {1,1,2,3}      92: {1,1,9}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y], y], UnsameQ@@#&];
Select[Range[100], Length[pofprop[prix[#]]]==0&]

Crossrefs

The non-proper version appears to be A382912, counted by A383710.

The non-proper complement appears to be A382913, counted by A383708.

The complement is A384321, counted by A384317.

These partitions are counted by A384348.

These are the positions of 0 in A384389.

The case of a unique proper choice is A384390, counted by A384319.

A048767 is the Look-and-Say transform, fixed points A048768.

A056239 adds up prime indices, row sums of A112798.

A179009 counts maximally refined strict partitions, ranks A383707.

A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.

Cf. A122111, A130091, A317142, A326080, A351294, A357982, A381454, A382525, A383706, A384320, A384322.

Sequence in context: A363227 A225730 A256389 * A384320 A225729 A035033

Adjacent sequences: A384346 A384347 A384348 * A384350 A384351 A384352

Keyword

nonn

Author

Gus Wiseman, Jun 03 2025

Status

approved