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proposed

The version for strict partitions appears to be {1,3}.

A340831/A340832 counts count factorizations with odd maximum/minimum.

Positions of nonzero terms in A340851, which counts these These factorizations are counted by A340851.

For example, 16000 has factorization 2*2*2*2*2*2*2*5*5*5, so is in the sequence.

For example, 24576 has three suitable factorizations:

Partitions of this type are counted by A340693 (A340606).

A340693 counts partitions of this type (A340606).

A143773 counts partitions whose parts are multiples of the number of parts, strict case A340830.

A340101 counts factorizations into odd factors (A066208), with , odd-length case A340102.

A340785 counts factorizations into even numbers, with even-length case A340786.

Cf. A050320, A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340609, A340654, A340655, A340827, ~A340828A340830.

allocated for Gus WisemanNumbers that can be factored in such a way that every factor is a divisor of the number of factors.

1, 4, 16, 27, 32, 64, 96, 128, 144, 192, 216, 256, 288, 324, 432, 486, 512, 576, 648, 729, 864, 972, 1024, 1296, 1458, 1728, 1944, 2048, 2560, 2592, 2916, 3125, 3888, 4096, 5120, 5184, 5832, 6144, 6400, 7776, 8192, 9216, 11664, 12288, 12800, 13824, 15552

1,2

Also numbers that can be factored in such a way that the length is divisible by the least common multiple.

The sequence of terms together with their prime indices begins:

1: {}

4: {1,1}

16: {1,1,1,1}

27: {2,2,2}

32: {1,1,1,1,1}

64: {1,1,1,1,1,1}

96: {1,1,1,1,1,2}

128: {1,1,1,1,1,1,1}

144: {1,1,1,1,2,2}

192: {1,1,1,1,1,1,2}

216: {1,1,1,2,2,2}

256: {1,1,1,1,1,1,1,1}

288: {1,1,1,1,1,2,2}

324: {1,1,2,2,2,2}

432: {1,1,1,1,2,2,2}

For example, 16000 has factorization 2*2*2*2*2*2*2*5*5*5, so is in the sequence.

For example, 24576 has factorizations:

(2*2*2*2*2*2*2*2*2*2*2*12)

(2*2*2*2*2*2*2*2*2*2*4*6)

(2*2*2*2*2*2*2*2*2*3*4*4)

so is in the sequence.

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

Select[Range[1000], Select[facs[#], And@@IntegerQ/@(Length[#]/#)&]!={}&]

Positions of nonzero terms in A340851, which counts these factorizations.

The reciprocal version is A340853.

The version for strict partitions appears to be {1,3}.

A143773 counts partitions whose parts are multiples of the number of parts, strict case A340830.

A320911 can be factored into squarefree semiprimes.

A340597 have an alt-balanced factorization.

A340656 lack a twice-balanced factorization, complement A340657.

A340693 counts partitions of this type (A340606).

- Factorizations -

A001055 counts factorizations, with strict case A045778.

A316439 counts factorizations by product and length.

A339846 counts factorizations of even length.

A339890 counts factorizations of odd length.

A340101 counts factorizations into odd factors (A066208), with odd-length case A340102.

A340653 counts balanced factorizations.

A340831/A340832 counts factorizations with odd maximum/minimum.

A340785 counts factorizations into even numbers, with even-length case A340786.

A340854 cannot be factored with odd least factor, complement A340855.

Cf. A050320, A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340609, A340654, A340655, A340827, ~A340828.

allocated

nonn

Gus Wiseman, Feb 04 2021

approved

editing

allocated for Gus Wiseman

allocated

approved