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A324460 Numbers m > 1 that have a strict s-decomposition.

%I A324460 #24 Oct 05 2024 18:11:24
%S A324460 45,96,225,325,405,576,637,640,891,1225,1377,1408,1536,1701,1729,2025,
%T A324460 2541,2821,3321,3751,3825,4225,4608,4961,6400,6517,6525,7381,7840,
%U A324460 8125,8281,9216,9537,9801,10625,10935,12025,12288,12825,12936,13125,13312,13357
%N A324460 Numbers m > 1 that have a strict s-decomposition.
%C A324460 The sequence contains the primary Carmichael numbers A324316.
%C A324460 The sequence is infinite. If f(x) counts such numbers m below x, then f(x) > 1/11 x^(1/3) - 1/3 for x >= 1.
%C A324460 A number m > 1 has a strict 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 A324460     m = g_1^e_1 * ... * g_n^e_n, where s_{g_k}(m) = g_k for all k,
%C A324460   and s_g(m) gives the sum of the base-g digits of m.
%C A324460 A term m has the following properties:
%C A324460 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 A324460 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 A324460 See Kellner 2019.
%H A324460 Bernd C. Kellner, <a href="/A324460/b324460.txt">Table of n, a(n) for n = 1..254</a>
%H A324460 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 A324460 Since 576 = 2^4 * 6^2 with s_2(576) = 2 and s_6(576) = 6, 576 is a member.
%t A324460 s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
%t A324460 HasDecompS[m_] := Module[{E0, EV, G, R, k, n, v},
%t A324460 If[m < 1 || !CompositeQ[m], Return[False]];
%t A324460 G = Select[Divisors[m], s[m, #] == # &];
%t A324460 n = Length[G]; If[n < 2, Return[False]];
%t A324460 E0 = Array[0 &, n]; EV = Array[v, n];
%t A324460 R = Solve[Product[G[[k]]^EV[[k]], {k, 1, n}] == m && EV >= E0, EV, Integers]; Return[R != {}]];
%t A324460 Select[Range[10^4], HasDecompS[#] &]
%Y A324460 Subsequences are A324316, A324458. Subsequence of A324459.
%Y A324460 Cf. A324455, A324456, A324457.
%K A324460 nonn,base
%O A324460 1,1
%A A324460 _Bernd C. Kellner_, Feb 28 2019