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A384330 Number of distinct subsets S of [n] such that for all 1 <= k <= n, there exist elements x,y in S (not necessarily distinct) such that x*y = 2k.

%I A384330 #26 May 26 2025 18:29:21
%S A384330 1,0,1,1,1,1,3,3,8,11,30,30,57,57,159,295,427,427,1033,1033,1973,3610,
%T A384330 10427,10427,20575,28731,83535,142793,273755,273755,549946,549946,
%U A384330 1245416,2289562,6665252,12386159,24210731,24210731,71150197,131657471,256115337,256115337
%N A384330 Number of distinct subsets S of [n] such that for all 1 <= k <= n, there exist elements x,y in S (not necessarily distinct) such that x*y = 2k.
%F A384330 a(p) = a(p-1) for odd prime p. - _Jinyuan Wang_, May 26 2025
%e A384330 a(6) = 3 because there are three sets that match the said condition: {1,2,3,4,5}, {1,2,4,5,6} and {1,2,3,4,5,6}.
%o A384330 (Python)
%o A384330 def a(n):
%o A384330     if n == 0: return 1
%o A384330     t = set(k << 1 for k in range(1, n+1))
%o A384330     c = 0
%o A384330     for i in range(1, (1 << n)+1, 2):
%o A384330         s = [j+1 for j in range(n) if (i >> j) & 1]
%o A384330         if len(s) == 0 or s[0] != 1: continue
%o A384330         ss = set(x * y for x in s for y in s if not (x & 1 and y & 1) )
%o A384330         if t.issubset(ss): c += 1
%o A384330     return c
%o A384330 print([a(n) for n in range(0,29)])
%Y A384330 Cf. A000079, A000225, A383968.
%K A384330 nonn
%O A384330 0,7
%A A384330 _Darío Clavijo_, May 26 2025
%E A384330 More terms from _Jinyuan Wang_, May 26 2025