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

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Number of condensed integer factorizations of n into unordered factors > 1, meaning it is 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 A368098, Factors instead of divisors: A368414, complement A368097A368413, unique A370645.

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

For partitions Subsets of this type: A370582 and A370636, complement A370583 and prime factors we have A370592, ranks A368100A370637.

The complement for partitions and prime factors is A370593, ranks A355529.

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

A006530 gives greatest prime factor, least A020639.

A027746 lists prime factors, A112798 indices, length A001222.

A355741, A355744, A355745 count choices of prime factors of prime indices.

Cf. ~A003963, A239312, A340596, A340653, ~A355535, A355739, ~A367867, A368110, A370592, A370803, A370804, A370805, `A370806, `A370807, `A370808.

allocated for Gus WisemanNumber of condensed integer factorizations of n into unordered factors > 1, meaning it is possible to choose a different divisor of each factor.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2

1,4

The a(36) = 7 factorizations: (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), (6*6), (36).

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 of this type are counted by A239312, ranks A368110.

For unlabeled multiset partitions we have A368098, complement A368097.

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

Partitions not of this type are counted by A370320, ranks A355740.

For partitions and prime factors we have A370592, ranks A368100.

The complement for partitions and prime factors is A370593, ranks A355529.

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

The complement is counted by A370813, partitions A370593, ranks A355529.

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.

A355741, A355744, A355745 count choices of prime factors of prime indices.

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

Cf. ~A003963, A239312, A340596, A340653, ~A355535, A355739, ~A367867, A368110, 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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