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

%I A195101 #31 Apr 10 2024 11:14:09
%S A195101 1,6,8,9,12,14,15,16,18,20,21,24,26,27,30,32,34,35,36,38,39,40,42,44,
%T A195101 45,48,50,51,52,54,56,57,60,62,63,64,66,68,70,72,74,75,76,78,80,81,84,
%U A195101 86,87,88,90,92,93,95,96,98,99,100,102,104,105,108,110,111
%N A195101 Solid numbers. Numbers m such that A005245(m) < A005245(k) + A005245(m-k) for 1<= k < m.
%C A195101 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.
%C A195101 The solid numbers  <= x appear to be  <= 0.6 * x.
%C A195101 We find many values where the minimum of A005245(k) + A005245(m-k) is not taken for k = 1. This is sequence A189123.
%C A195101 The first value of m needing k = 6 is 21080618, the first k = 8 is 385159320, the first with k = 9 is 3679353584.
%C A195101 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).
%C A195101 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
%H A195101 Juan Arias-de-Reyna, <a href="/A195101/b195101.txt">Table of n, a(n) for n = 1..10000</a>
%H A195101 H. Altman and J. Zelinsky, <a href="http://arxiv.org/abs/1207.4841">Numbers with integer complexity close to the lower bound</a>, Integers, 12 (2012) 1093-1125 (article where sequence is first introduced).
%H A195101 Juan Arias de Reyna, <a href="https://arxiv.org/abs/2302.06224">Complexity of natural numbers and arithmetic compact sets</a>, arXiv:2302.06224 [math.NT], 2023.
%H A195101 Juan Arias de Reyna, <a href="http://math.colgate.edu/~integers/y21/y21.pdf">Arithmetical Self-Similar Compact Sets</a>, Integers (2024) Vol. 24, A21. See p. 13.
%e A195101 m = 8 is a term of the sequence because
%e A195101 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;
%e A195101 A005245(8) < A005245(4) + A005245(4)=4+4.
%e A195101 m = 7 is not a term of the sequence because A005245(7) = 6 = A005245(6) + A005245(1) = 5 + 1.
%t A195101 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 *)
%Y A195101 Cf. A005245, A189123.
%K A195101 nonn
%O A195101 1,2
%A A195101 _Juan Arias-de-Reyna_, Sep 09 2011
%E A195101 Name and comments change using "solid numbers" notation by _Juan Arias-de-Reyna_, Jan 09 2014