Max Alekseyev, <a href="/A370816/b370816_1.txt">Table of n, a(n) for n = 1..10000</a>

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Max Alekseyev, <a href="/A370816/b370816_1.txt">Table of n, a(n) for n = 1..10000</a>

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For just prime factors we have A370817, for partitions A370809.

For just prime factors we have A370817.

A000041 counts integer partitions, strict A000009.

A006530 gives greatest prime factor, least A020639.

A027746 lists prime factors, A112798 indices, length A001222.

A355741 chooses prime factors of prime indices, variations A355744, A355745.

A370593 counts non-choosable partitions, complement A370592, see also A370594, `A370807.

A370812 counts ways to choose divisors of prime indices, non-strict A355733.

Cf. A000792, `A048249, `A066739, A319055, A319057, A355737, A355739, A355740, `A355749, ~A367771, A355741, A368110, ~A370348, ~A370585, `A370595, `A370803, ~A370810.

allocated for Gus WisemanGreatest number of multisets that can be obtained by choosing a divisor of each factor in an integer factorization of n into unordered factors > 1.

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 7, 2, 4, 4, 7, 2, 7, 2, 7, 4, 4, 2, 11, 3, 4, 5, 7, 2, 8, 2, 10, 4, 4, 4, 12, 2, 4, 4, 11, 2, 8, 2, 7, 7, 4, 2, 17, 3, 7, 4, 7, 2, 11, 4, 11, 4, 4, 2, 15, 2, 4, 7, 14, 4, 8, 2, 7, 4, 8, 2, 20, 2, 4, 7, 7, 4, 8, 2, 17, 7, 4, 2

1,2

For the factorizations of 12 we have the following choices:

(2*2*3): {{1,1,1},{1,1,2},{1,1,3},{1,2,2},{1,2,3},{2,2,3}}

(2*6): {{1,1},{1,2},{1,3},{1,6},{2,2},{2,3},{2,6}}

(3*4): {{1,1},{1,2},{1,3},{1,4},{2,3},{3,4}}

(12): {{1},{2},{3},{4},{6},{12}}

So a(12) = 7.

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

Table[Max[Length[Union[Sort/@Tuples[Divisors/@#]]]&/@facs[n]], {n, 100}]

For just prime factors we have A370817, for partitions A370809.

The version for partitions is A370808, for just prime factors A370809.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

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 chooses prime factors of prime indices, variations A355744, A355745.

A368413 counts non-choosable factorizations, complement A368414.

A370593 counts non-choosable partitions, complement A370592, see also A370594, `A370807.

A370812 counts ways to choose divisors of prime indices, non-strict A355733.

A370813 counts non-divisor-choosable factorizations, complement A370814.

Cf. A000792, `A048249, `A066739, A319055, A319057, A355737, A355739, A355740, `A355749, ~A367771, A368110, ~A370348, ~A370585, `A370595, `A370803, ~A370810.

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Gus Wiseman, Mar 06 2024

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allocated for Gus Wiseman

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