A348083 - OEIS

%I #18 Oct 05 2021 22:47:54

%S 1,1,1,1,2,3,2,6,6,8,13,18,21,35,45,61,90,121,162,241,323,450,638,865,

%T 1233,1698,2349,3315,4592,6382,8970,12421,17351,24320,33714,47218,

%U 65978,91784,128177,179807,249689,349549,489341,681468,953769,1334490,1860641,2606043,3643618,5086481,7124229,9960420

%N Number of positive numbers that can be built with n ones using +, -, and *, and require at least n ones.

%C a(n+1)/a(n) appears from the values through n=63 to be oscillating in a narrowing range around 7/5.

%H Glen Whitney, <a href="/A348083/b348083.txt">Table of n, a(n) for n = 1..64</a> (This file produced with the same C code that generated A348069, simply with division eliminated from the loop over operators.)

%F a(n) = |{k : A091333(k) = n}|.

%e For n=5, there are two numbers whose minimal expression using 1,+,-, and * uses five ones: 5 = 1+1+1+1+1 and 6 = (1+1)*(1+1+1), so a(5) = 2.

%e For n=10, there are eight numbers whose minimal expression uses ten ones: 22 = 3(2*3+1)+1, 23 = 2*2*2*3-1, 25 = 5*5, 26 = 3*3*3-1, 28 = 3*3*3+1, 30 = 2*3*5, 32 = 2*2*2*2*2, and 36 = 2*2*3*3. We use numbers k=1..5 in these expressions because each takes k ones to express. Note that n=10 is also the least n for which a(n) differs from A005421(n), which counts the solutions to A005245(k) = n.

%o (Python)

%o from functools import cache

%o @cache

%o def f(m):

%o     if m == 0: return set()

%o     if m == 1: return {1}

%o     out = set()

%o     for j in range(1, m//2+1):

%o         for x in f(j):

%o             for y in f(m-j):

%o                 out.update([x + y, x * y])

%o                 if x != y: out.add(abs(x-y))

%o     return list(out)

%o def a(n): return len(f(n)) - len(f(n-1))

%o print([a(n) for n in range(1, 33)]) # _Michael S. Branicky_, Sep 27 2021

%Y Cf. A005421, A091333, A005245.

%K nonn

%O 1,5

%A _Glen Whitney_, Sep 27 2021