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A370812 chooses a different divisor of each prime index, non-strict A355733.

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Number of non-condensed integer factorizations of n into unordered factors > 1, meaning it is not possible to choose a different divisor of each factor.

A multiset is condensed iff it is possible to choose a different divisor of each element.

For unlabeled multiset partitions we have A368097, Factors instead of divisors: A368413, complement A368098A368414, unique A370645.

For prime factors we have A368413, complement A368414, unique A370645.

The Subsets of this type: A370583 and A370637, complement for partitions A370582 and prime factors is A370592, ranks A368100A370636.

For partitions and prime factors we have A370593, ranks A355529.

Subsets of this type are counted by A370637, complement A370636.

A006530 gives greatest prime factor, least A020639.

A027746 lists prime factors, A112798 indices, length A001222.

Cf. ~A003963, A239312, A340596, A340653, `A355535, A355529, A355739, A355741, A368110, `A370583, ~A370584, `A370594, `A370803, A370804, `A370805, `A370806, A370807, `A370808.

allocated for Gus WisemanNumber of non-condensed integer factorizations of n into unordered factors > 1, meaning it is not possible to choose a different divisor of each factor.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0

1,32

The a(96) = 4 factorizations: (2*2*2*2*2*3), (2*2*2*2*6), (2*2*2*3*4), (2*2*2*12).

facs[n_]:=If[n<=1, {{}}, Join @@ Table[Map[Prepend[#, d]&, Select[facs[n/d], Min @@ #>=d&]], {d, Rest[Divisors[n]]}]];

Table[Length[Select[facs[n], Length[Select[Tuples[Divisors /@ #], UnsameQ@@#&]]==0&]], {n, 100}]

Partitions not of this type are counted by A239312, ranks A368110.

For unlabeled multiset partitions we have A368097, complement A368098.

For prime factors we have A368413, complement A368414, unique A370645.

Partitions of this type are counted by A370320, ranks A355740.

The complement for partitions and prime factors is A370592, ranks A368100.

For partitions and prime factors we have A370593, ranks A355529.

Subsets of this type are counted by A370637, complement A370636.

The complement is counted by A370814, partitions A370592, ranks A368100.

For a unique choice we have A370815, partitions A370595, ranks A370810.

A000005 counts divisors.

A001055 counts factorizations, strict A045778.

A006530 gives greatest prime factor, least A020639.

A027746 lists prime factors, A112798 indices, length A001222.

A355731 counts choices of a divisor of each prime index, firsts A355732.

A370812 chooses a different divisor of each prime index, non-strict A355733.

Cf. ~A003963, A239312, A340596, A340653, `A355535, A355739, A355741, A368110, `A370583, ~A370584, `A370594, `A370803, A370804, `A370805, `A370806, A370807, `A370808.

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Gus Wiseman, Mar 04 2024

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allocated for Gus Wiseman

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