login
Numbers m > 1 that have a strict s-decomposition.
8

%I #24 Oct 05 2024 18:11:24

%S 45,96,225,325,405,576,637,640,891,1225,1377,1408,1536,1701,1729,2025,

%T 2541,2821,3321,3751,3825,4225,4608,4961,6400,6517,6525,7381,7840,

%U 8125,8281,9216,9537,9801,10625,10935,12025,12288,12825,12936,13125,13312,13357

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

%C The sequence contains the primary Carmichael numbers A324316.

%C 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 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 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="/A324460/b324460.txt">Table of n, a(n) for n = 1..254</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 576 = 2^4 * 6^2 with s_2(576) = 2 and s_6(576) = 6, 576 is a member.

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

%t HasDecompS[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[10^4], HasDecompS[#] &]

%Y Subsequences are A324316, A324458. Subsequence of A324459.

%Y Cf. A324455, A324456, A324457.

%K nonn,base

%O 1,1

%A _Bernd C. Kellner_, Feb 28 2019