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

24, 45, 48, 72, 96, 120, 144, 189, 192, 216, 224, 225, 231, 240, 280, 288, 315, 320, 325, 336, 352, 360, 378, 384, 405, 432, 450, 480, 525, 540, 560, 561, 567, 576, 594, 600, 637, 640, 648, 672, 704, 720, 768, 792, 819, 825, 832, 850, 864, 891, 896, 924, 945

Offset: 1

Author

Bernd C. Kellner, Feb 28 2019

Keywords

• nonn
• base

Comments

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

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

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

Links

• Bernd C. Kellner, Table of n, a(n) for n = 1..532
• Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.

Crossrefs

Subsequences are A002997, A324457, A324458, A324460.

Cf. A324455, A324456.

Programs

Mathematica

s[n_, g_] := If[n < 1 || g < 2, 0, Plus @@ IntegerDigits[n, g]];
HasDecomp[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[10^3], HasDecomp[#] &]