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

Author

Amiram Eldar, Nov 11 2021

Keywords

• nonn

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).

Examples

			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.
		

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

Crossrefs

Cf. A002808, A050370.

Subsequence of A025487.

Programs

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