The On-Line Encyclopedia of Integer Sequences (OEIS)

A340852 revision #2

A340852

Numbers 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

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OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..47.

EXAMPLE

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.

MATHEMATICA

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[#]/#)&]!={}&]

CROSSREFS

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.

Sequence in context: A050707 A134330 A046346 * A328415 A097764 A227993

Adjacent sequences: A340849 A340850 A340851 * A340853 A340854 A340855

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 04 2021

STATUS

editing