A195101 Solid numbers. Numbers m such that A005245(m) < A005245(k) + A005245(m-k) for 1<= k < m.

1, 6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 26, 27, 30, 32, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 57, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 84, 86, 87, 88, 90, 92, 93, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 111

Offset: 1

Author

Juan Arias-de-Reyna, Sep 09 2011

Keywords

• nonn

Comments

It is useful for computing A005245(m). To compute min_k A005245(k) + A005245(m-k) we only need to check the cases in which k is a solid number.

The solid numbers <= x appear to be <= 0.6 * x.

We find many values where the minimum of A005245(k) + A005245(m-k) is not taken for k = 1. This is sequence A189123.

The first value of m needing k = 6 is 21080618, the first k = 8 is 385159320, the first with k = 9 is 3679353584.

Conjecture that for every solid number m > 1 there is some number n such that A005245(n) = A005245(m)+A005245(n-m) and such that for every representation as a product n = u*v with u, v >= 2 or every 1 < = k < m, we have A005245(n) < A005245(u)+A005245(v) and A005245(n) < A005245(k) + A005245(n-k).

The solid numbers are infinite. Proof by H. Altman, mentioned in link. For n>1, 3^n is a solid number. If 3^n=a+b with 3n=||a||+||b||, then 3log_3(a)+3log_3(b)<=3n, and so ab<=3^n=a+b. So either a=b=2 (impossible), or a=1 or b=1. So suppose a=1. Then b=3^n-1. But since n>1 we have 3^n-1>(3/4)3^n>=E(3n-1) and thus ||3^n-1||>=3n, ||a||+||b||>=3n+1, contradiction. - Juan Arias-de-Reyna, Jan 09 2014

Examples

			m = 8 is a term of the sequence because
A005245(8) = 6 < A005245(7) + A005245(1)=6+1; A005245(8) < A005245(6) + A005245(2)=5+2; A005245(8)  < A005245(5) + A005245(3)=5+3;
A005245(8) < A005245(4) + A005245(4)=4+4.
m = 7 is not a term of the sequence because A005245(7) = 6 = A005245(6) + A005245(1) = 5 + 1.
		

Links

• Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
• H. Altman and J. Zelinsky, Numbers with integer complexity close to the lower bound, Integers, 12 (2012) 1093-1125 (article where sequence is first introduced).
• Juan Arias de Reyna, Complexity of natural numbers and arithmetic compact sets, arXiv:2302.06224 [math.NT], 2023.
• Juan Arias de Reyna, Arithmetical Self-Similar Compact Sets, Integers (2024) Vol. 24, A21. See p. 13.

Crossrefs

Cf. A005245, A189123.

Programs

Mathematica

nn = 200; a5245[n_] :=  a5245[n] = If[n == 1, 1, Min[Sequence @@ Table[a5245[i] + a5245[n - i], {i, 1, n/2}], Sequence @@ Table[a5245[d] + a5245[n/d], {d, Divisors[n]~Complement~{1, n}}]]]; t = Table[a5245[n], {n, nn}]; Select[Range[nn], And @@ Table[t[[#]] < t[[k]] + t[[# - k]], {k, # - 1}] &] (* T. D. Noe, Apr 09 2014 *)

Extensions

Name and comments change using "solid numbers" notation by Juan Arias-de-Reyna, Jan 09 2014