A384393 Heinz numbers of integer partitions with more than one proper way to choose disjoint strict partitions of each part.

11, 13, 17, 19, 23, 25, 29, 31, 34, 37, 38, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134

Offset

1,1

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..61.

Example

The prime indices of 275 are {3,3,5}, with a total of 2 proper choices: ((3),(2,1),(5)) and ((2,1),(3),(5)), so 275 is in the sequence.
The terms together with their prime indices begin:
    11: {5}      51: {2,7}      82: {1,13}
    13: {6}      53: {16}       83: {23}
    17: {7}      55: {3,5}      85: {3,7}
    19: {8}      57: {2,8}      86: {1,14}
    23: {9}      58: {1,10}     87: {2,10}
    25: {3,3}    59: {17}       89: {24}
    29: {10}     61: {18}       91: {4,6}
    31: {11}     62: {1,11}     93: {2,11}
    34: {1,7}    65: {3,6}      94: {1,15}
    37: {12}     67: {19}       95: {3,8}
    38: {1,8}    69: {2,9}      97: {25}
    41: {13}     71: {20}      101: {26}
    43: {14}     73: {21}      103: {27}
    46: {1,9}    74: {1,12}    106: {1,16}
    47: {15}     77: {4,5}     107: {28}
    49: {4,4}    79: {22}      109: {29}

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[#]]]>1&]

Crossrefs

Without "proper" we get A384321 (strict A384322), counted by A384317 (strict A384318).

The case of no choices is A384349, counted by A384348.

These are positions of terms > 1 in A384389.

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

Partitions of this type are counted by A384395.

A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A239455 counts Look-and-Say partitions, ranks A351294 or A381432.

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

A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.

Cf. A382912, A383710, A382913, A383708, A383533.

Cf. A179009, A357982, A381454, A382525, A383706, A383707, A384320, A384323.

Sequence in context: A191599 A084374 A063193 * A056758 A392375 A322273

Adjacent sequences: A384390 A384391 A384392 * A384394 A384395 A384396

Keyword

nonn

Author

Gus Wiseman, Jun 02 2025

Status

approved