A394865 - OEIS

%I #45 Jun 17 2026 07:34:57

%S 1,1,1,2,3,3,4,4,4,4,5,5,6,6,6,7,8,8,9,10,10,10,11,11,12,12,12,13,14,

%T 14,15,15,15,15,16,16,17,17,17,17,18,18,19,20,20,20,21,21,22,22,22,23,

%U 24,24,25,25,25,25,26,26,27,27,27,28,29,29,30,31,31,31

%N a(n) is the size of the largest subset A of {1, ..., n} such that x does not divide 2*y for each pair of distinct elements x, y in A.

%C a(n) is the number of terms of A084087 not exceeding n.

%C The construction of the largest subset is to choose all numbers n/3 < m <= n such that the 2-valuation of m is even.

%H Paolo Xausa, <a href="/A394865/b394865.txt">Table of n, a(n) for n = 1..10000</a>

%F Limit_{n->oo} a(n)/n = 4/9.

%F a(n) = A050292(n) - A050292(floor(n/3)). - _David Radcliffe_, Jun 13 2026

%e For n = 7, the largest such subset is {3, 4, 5, 7}, so a(7) = 4.

%t A050292[n_] := A050292[n] = If[n == 0, 0, n - A050292[Quotient[n, 2]]];

%t A394865[n_] := A050292[n] - A050292[Quotient[n, 3]];

%t Array[A394865, 100] (* Paolo Xausa, Jun 17 2026 *)

%t (* Alternative: *)

%t Accumulate[Array[Boole[!Divisible[#, 3] && EvenQ[IntegerExponent[#, 2]]] &, 100]] (* _Paolo Xausa_, Jun 17 2026 *)

%o (PARI) lista(nn) = {my(cnt = 0); for(n = 1, nn, if(gcd(n, 3) == 1 && valuation(n, 2) % 2 == 0, cnt++); if(n > 1, print1(", ")); print1(cnt); ); }

%Y Cf. A050292, A084087.

%K nonn,easy

%O 1,4

%A _Yifan Xie_, Jun 12 2026