%I #18 Jul 10 2020 03:43:19
%S 1,1,2,2,2,2,3,3,2,3,4,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,4,5,5,4,
%T 5,6,5,5,6,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,5,6,6,4,6,6,5,6,6,6,6,6,7,6,
%U 6,8,7,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,6,7,7,6,7
%N a(n) = a(floor((n-1)/a(n-1))) + a(floor((n-2)/a(n-2))) with a(1) = a(2) = 1.
%C This sequence is a_1(n) where a_i(n) = Sum_{k=1..i+1} a_i(floor((n-k)/a_i(n-k))) with a_i(n) = 1 for n <= i+1.
%C Conjecture: This sequence hits every positive integer.
%H Altug Alkan, <a href="/A335898/b335898.txt">Table of n, a(n) for n = 1..10000</a>
%t a[1] = a[2] = 1; a[n_] := a[n] = a[Floor[(n-1)/a[n-1]]] + a[Floor[(n-2)/a[n-2]]]; Array[a, 100] (* _Amiram Eldar_, Jun 29 2020 *)
%o (PARI) a=vector(10^2); a[1]=a[2]=1; for(n=3, #a, a[n]=a[(n-1)\a[n-1]]+a[(n-2)\a[n-2]]); a
%Y Cf. A130535, A283207.
%K nonn,look,easy
%O 1,3
%A _Altug Alkan_, following a suggestion from _Andrew R. Booker_, Jun 29 2020