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A324459 Numbers m > 1 that have an s-decomposition.

%I A324459 #24 Oct 05 2024 18:12:22
%S A324459 24,45,48,72,96,120,144,189,192,216,224,225,231,240,280,288,315,320,
%T A324459 325,336,352,360,378,384,405,432,450,480,525,540,560,561,567,576,594,
%U A324459 600,637,640,648,672,704,720,768,792,819,825,832,850,864,891,896,924,945
%N A324459 Numbers m > 1 that have an s-decomposition.
%C A324459 The sequence is infinite, since it contains A324460 and the Carmichael numbers A002997.
%C A324459 A number m > 1 has an s-decomposition if there exists a decomposition in n proper factors g_k with exponents e_k >= 1 (the factors g_k being strictly increasing but not necessarily coprime) such that
%C A324459     m = g_1^e_1 * ... * g_n^e_n, where s_{g_k}(m) >= g_k for all k,
%C A324459   and s_g(m) gives the sum of the base-g digits of m.
%C A324459 A term m has the following properties:
%C A324459 m must have at least 2 factors g_k. If m = g_1^e_1 * g_2^e_2 with exactly two factors, then e_1 + e_2 >= 3.
%C A324459 Each factor g_k of m satisfies the inequalities 1 < g_k < m^(1/(ord_{g_k}(m)+1)) <= sqrt(m), where ord_g(m) gives the maximum exponent e such that g^e divides m.
%C A324459 See Kellner 2019.
%H A324459 Bernd C. Kellner, <a href="/A324459/b324459.txt">Table of n, a(n) for n = 1..532</a>
%H A324459 Bernd C. Kellner, <a href="https://doi.org/10.5281/zenodo.10963985">On primary Carmichael numbers</a>, Integers 22 (2022), Article #A38, 39 pp.; arXiv:<a href="https://arxiv.org/abs/1902.11283">1902.11283</a> [math.NT], 2019.
%e A324459 Since 225 = 5^2 * 9 with s_5(225) = 5 and s_9(225) = 9, 225 is a member.
%t A324459 s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
%t A324459 HasDecomp[m_] := Module[{E0, EV, G, R, k, n, v},
%t A324459 If[m < 1 || !CompositeQ[m], Return[False]];
%t A324459 G = Select[Divisors[m], s[m, #] >= # &];
%t A324459 n = Length[G]; If[n < 2, Return[False]];
%t A324459 E0 = Array[0 &, n]; EV = Array[v, n];
%t A324459 R = Solve[Product[G[[k]]^EV[[k]], {k, 1, n}] == m && EV >= E0, EV, Integers]; Return[R != {}]];
%t A324459 Select[Range[10^3], HasDecomp[#] &]
%Y A324459 Subsequences are A002997, A324457, A324458, A324460.
%Y A324459 Cf. A324455, A324456.
%K A324459 nonn,base
%O A324459 1,1
%A A324459 _Bernd C. Kellner_, Feb 28 2019