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Also BII-numbers (see A326031) of set-systems with no singleton edges. For example, the sequence of such set-systems together with their BII-numbers begins:

0: {}

4: {{1,2}}

16: {{1,3}}

20: {{1,2},{1,3}}

32: {{2,3}}

36: {{1,2},{2,3}}

48: {{1,3},{2,3}}

52: {{1,2},{1,3},{2,3}}

64: {{1,2,3}}

68: {{1,2},{1,2,3}}

80: {{1,3},{1,2,3}}

84: {{1,2},{1,3},{1,2,3}}

96: {{2,3},{1,2,3}}

100: {{1,2},{2,3},{1,2,3}}

112: {{1,3},{2,3},{1,2,3}}

116: {{1,2},{1,3},{2,3},{1,2,3}}

256: {{1,4}}

260: {{1,2},{1,4}}

272: {{1,3},{1,4}}

276: {{1,2},{1,3},{1,4}}

Here a half-loop is an edge with only one vertex, to be distinguished from a full loop, which has two equal vertices.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

Also products of distinct primes whose prime indices are either themselves prime or a squarefree semiprime, and whose prime indices together cover an initial interval of positive integers. A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. 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 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, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges. A set-system is uniform if all edges have the same size, and regular if all vertices appear the same number of times.