A349223 Numbers k with a record number of ways to factor k into composite factors.

1, 4, 16, 36, 64, 96, 144, 240, 288, 360, 432, 480, 576, 720, 864, 960, 1152, 1440, 2160, 2880, 4320, 5040, 5760, 7200, 8640, 10080, 11520, 12960, 14400, 15120, 17280, 20160, 25920, 28800, 30240, 34560, 40320, 50400, 51840, 60480, 80640, 86400, 90720, 100800, 120960

Offset

1,2

Comments

Except for the term 4, the indices of records of A050370.

The corresponding record values are 0, 1, 2, 3, 4, 5, 8, 9, 11, 12, 13, 14, ... (see the link for more values).

Links

Amiram Eldar, Table of n, a(n) for n = 1..200

Amiram Eldar, Table of n, a(n) and record values for n = 1..200

Example

a(1) = 1 since 1 has 0 ways to be factored into composite factors.
a(2) = 4 since 4 is composite and has 1 presentation as a product of composite numbers: 4.
a(3) = 16 since 16 has 2 presentations as a product of composite numbers: 4 * 4 and 16.
a(4) = 36 since 36 has 3 presentations as a product of composite numbers: 6 * 6, 4 * 9 and 36.

Mathematica

f[_, 1] = f[1, _] = 1; f[n_, m_] := f[n, m] = DivisorSum[n, f[n/#, #] &, 1 < # <= m &]; g[n_] := DivisorSum[n, MoebiusMu[n/#]*f[#, #] &]; gm = 0; seq = {1}; Do[g1 = g[n]; If[g1 > gm, gm = g1; AppendTo[seq, n]], {n, 2, 10^5}]; seq

Crossrefs

Cf. A002808, A050370.

Subsequence of A025487.

Sequence in context: A044065 A281795 A063540 * A055808 A016742 A221285

Adjacent sequences: A349220 A349221 A349222 * A349224 A349225 A349226

Keyword

nonn

Author

Amiram Eldar, Nov 11 2021

Status

approved