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%I #6 Nov 20 2019 09:34:09
%S 0,1,2,8,4,3,128,16,32768,9,5,2147483648,256,32,129,10,
%T 9223372036854775808,6,170141183460469231731687303715884105728,512,
%U 65536,57896044618658097711785492504343953926634992332820282019728792003956564819968,130,17,32769,4294967296
%N BII-number of the set-system whose MM-number is A329629(n).
%C 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 set-system (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, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
%C 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}}.
%F A326031(a(n)) = A302242(A329629(n)).
%e The sequence of all set-systems together with their MM-numbers and BII-numbers begins:
%e {}: 1 ~ 0
%e {{1}}: 3 ~ 1
%e {{2}}: 5 ~ 2
%e {{3}}: 11 ~ 8
%e {{1,2}}: 13 ~ 4
%e {{1},{2}}: 15 ~ 3
%e {{4}}: 17 ~ 128
%e {{1,3}}: 29 ~ 16
%e {{5}}: 31 ~ 32768
%e {{1},{3}}: 33 ~ 9
%e {{1},{1,2}}: 39 ~ 5
%e {{6}}: 41 ~ 2147483648
%e {{1,4}}: 43 ~ 256
%e {{2,3}}: 47 ~ 32
%e {{1},{4}}: 51 ~ 129
%e {{2},{3}}: 55 ~ 10
%e {{7}}: 59 ~ 9223372036854775808
%e {{2},{1,2}}: 65 ~ 6
%e {{8}}: 67 ~ 170141183460469231731687303715884105728
%e {{2,4}}: 73 ~ 512
%t fbi[q_]:=If[q=={},0,Total[2^q]/2];
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t das=Select[Range[100],OddQ[#]&&SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&];
%t Table[fbi[fbi/@primeMS/@primeMS[n]],{n,das}]
%Y MM-numbers of set-systems are A329629.
%Y Cf. A000120, A005117, A048793, A056239, A070939, A112798, A302242, A302494, A326031, A329557.
%Y Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).
%Y Classes of BII-numbers: A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326752 (hypertrees), A326754 (covers).
%K nonn
%O 1,3
%A _Gus Wiseman_, Nov 19 2019