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Numbers k with a record number of ways to factor k into composite factors.
1

%I #15 Nov 12 2021 04:36:40

%S 1,4,16,36,64,96,144,240,288,360,432,480,576,720,864,960,1152,1440,

%T 2160,2880,4320,5040,5760,7200,8640,10080,11520,12960,14400,15120,

%U 17280,20160,25920,28800,30240,34560,40320,50400,51840,60480,80640,86400,90720,100800,120960

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

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

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

%H Amiram Eldar, <a href="/A349223/b349223.txt">Table of n, a(n) for n = 1..200</a>

%H Amiram Eldar, <a href="/A349223/a349223.txt">Table of n, a(n) and record values for n = 1..200</a>

%e a(1) = 1 since 1 has 0 ways to be factored into composite factors.

%e a(2) = 4 since 4 is composite and has 1 presentation as a product of composite numbers: 4.

%e a(3) = 16 since 16 has 2 presentations as a product of composite numbers: 4 * 4 and 16.

%e a(4) = 36 since 36 has 3 presentations as a product of composite numbers: 6 * 6, 4 * 9 and 36.

%t 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

%Y Cf. A002808, A050370.

%Y Subsequence of A025487.

%K nonn

%O 1,2

%A _Amiram Eldar_, Nov 11 2021