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

45, 96, 225, 325, 405, 576, 637, 640, 891, 1225, 1377, 1408, 1536, 1701, 1729, 2025, 2541, 2821, 3321, 3751, 3825, 4225, 4608, 4961, 6400, 6517, 6525, 7381, 7840, 8125, 8281, 9216, 9537, 9801, 10625, 10935, 12025, 12288, 12825, 12936, 13125, 13312, 13357

Offset: 1

Author

Bernd C. Kellner, Feb 28 2019

Keywords

• nonn
• base

Comments

The sequence contains the primary Carmichael numbers A324316.

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.

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

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

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

A term m has the following properties:

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.

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.

See Kellner 2019.

Examples

			Since 576 = 2^4 * 6^2 with s_2(576) = 2 and s_6(576) = 6, 576 is a member.
		

Links

• Bernd C. Kellner, Table of n, a(n) for n = 1..254
• Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.

Crossrefs

Subsequences are A324316, A324458. Subsequence of A324459.

Cf. A324455, A324456, A324457.

Programs

Mathematica

s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
HasDecompS[m_] := Module[{E0, EV, G, R, k, n, v},
If[m < 1 || !CompositeQ[m], Return[False]];
G = Select[Divisors[m], s[m, #] == # &];
n = Length[G]; If[n < 2, Return[False]];
E0 = Array[0 &, n]; EV = Array[v, n];
R = Solve[Product[G[[k]]^EV[[k]], {k, 1, n}] == m && EV >= E0, EV, Integers]; Return[R != {}]];
Select[Range[10^4], HasDecompS[#] &]