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Numbers m > 1 that have an s-decomposition.
6

%I #24 Oct 05 2024 18:12:22

%S 24,45,48,72,96,120,144,189,192,216,224,225,231,240,280,288,315,320,

%T 325,336,352,360,378,384,405,432,450,480,525,540,560,561,567,576,594,

%U 600,637,640,648,672,704,720,768,792,819,825,832,850,864,891,896,924,945

%N Numbers m > 1 that have an s-decomposition.

%C The sequence is infinite, since it contains A324460 and the Carmichael numbers A002997.

%C 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 m = g_1^e_1 * ... * g_n^e_n, where s_{g_k}(m) >= g_k for all k,

%C and s_g(m) gives the sum of the base-g digits of m.

%C A term m has the following properties:

%C 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 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 See Kellner 2019.

%H Bernd C. Kellner, <a href="/A324459/b324459.txt">Table of n, a(n) for n = 1..532</a>

%H 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 Since 225 = 5^2 * 9 with s_5(225) = 5 and s_9(225) = 9, 225 is a member.

%t s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];

%t HasDecomp[m_] := Module[{E0, EV, G, R, k, n, v},

%t If[m < 1 || !CompositeQ[m], Return[False]];

%t G = Select[Divisors[m], s[m, #] >= # &];

%t n = Length[G]; If[n < 2, Return[False]];

%t E0 = Array[0 &, n]; EV = Array[v, n];

%t R = Solve[Product[G[[k]]^EV[[k]], {k, 1, n}] == m && EV >= E0, EV, Integers]; Return[R != {}]];

%t Select[Range[10^3], HasDecomp[#] &]

%Y Subsequences are A002997, A324457, A324458, A324460.

%Y Cf. A324455, A324456.

%K nonn,base

%O 1,1

%A _Bernd C. Kellner_, Feb 28 2019