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%I A326031 #18 Jun 08 2024 01:48:17
%S A326031 0,1,1,2,2,3,3,4,1,2,2,3,3,4,4,5,2,3,3,4,4,5,5,6,3,4,4,5,5,6,6,7,2,3,
%T A326031 3,4,4,5,5,6,3,4,4,5,5,6,6,7,4,5,5,6,6,7,7,8,5,6,6,7,7,8,8,9,3,4,4,5,
%U A326031 5,6,6,7,4,5,5,6,6,7,7,8,5,6,6,7,7,8,8,9
%N A326031 Weight of the set-system with BII-number n.
%C A326031 A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets of positive integers has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, it follows that the BII-number of {{2},{1,3}} is 18. The weight of a set-system is the sum of sizes of its elements (sometimes called its edges).
%H A326031 John Tyler Rascoe, <a href="/A326031/b326031.txt">Table of n, a(n) for n = 0..8192</a>
%F A326031 a(2^x + ... + 2^z) = w(x + 1) + ... + w(z + 1), where x...z are distinct nonnegative integers and w = A000120. For example, a(6) = a(2^2 + 2^1) = w(3) + w(2) = 3.
%e A326031 The sequence of set-systems together with their BII-numbers begins:
%e A326031    0: {}
%e A326031    1: {{1}}
%e A326031    2: {{2}}
%e A326031    3: {{1},{2}}
%e A326031    4: {{1,2}}
%e A326031    5: {{1},{1,2}}
%e A326031    6: {{2},{1,2}}
%e A326031    7: {{1},{2},{1,2}}
%e A326031    8: {{3}}
%e A326031    9: {{1},{3}}
%e A326031   10: {{2},{3}}
%e A326031   11: {{1},{2},{3}}
%e A326031   12: {{1,2},{3}}
%e A326031   13: {{1},{1,2},{3}}
%e A326031   14: {{2},{1,2},{3}}
%e A326031   15: {{1},{2},{1,2},{3}}
%e A326031   16: {{1,3}}
%e A326031   17: {{1},{1,3}}
%e A326031   18: {{2},{1,3}}
%e A326031   19: {{1},{2},{1,3}}
%e A326031   20: {{1,2},{1,3}}
%t A326031 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326031 Table[Length[Join@@bpe/@bpe[n]],{n,0,100}]
%o A326031 (Python)
%o A326031 def bin_i(n): #binary indices
%o A326031     return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
%o A326031 def A326031(n): return sum(i.bit_count() for i in bin_i(n)) # _John Tyler Rascoe_, Jun 08 2024
%Y A326031 Cf. A000120, A029931, A048793, A061775, A070939, A072639, A116549, A302242, A305830, A326701, A326702, A326703, A326704.
%K A326031 nonn,base
%O A326031 0,4
%A A326031 _Gus Wiseman_, Jul 20 2019

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