cp's OEIS Frontend

A371450 MM-number of the set-system with BII-number n.

%I A371450 #6 Apr 02 2024 08:54:06
%S A371450 1,3,5,15,13,39,65,195,11,33,55,165,143,429,715,2145,29,87,145,435,
%T A371450 377,1131,1885,5655,319,957,1595,4785,4147,12441,20735,62205,47,141,
%U A371450 235,705,611,1833,3055,9165,517,1551,2585,7755,6721,20163,33605,100815,1363,4089
%N A371450 MM-number of the set-system with BII-number n.
%C A371450 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}}.
%C A371450 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.
%e A371450 The set-system with BII-number 30 is {{2},{1,2},{3},{1,3}} with MM-number prime(3) * prime(6) * prime(5) * prime(10) = 20735.
%e A371450 The terms together with their prime indices and binary indices of prime indices begin:
%e A371450      1 -> {}        -> {}
%e A371450      3 -> {2}       -> {{1}}
%e A371450      5 -> {3}       -> {{2}}
%e A371450     15 -> {2,3}     -> {{1},{2}}
%e A371450     13 -> {6}       -> {{1,2}}
%e A371450     39 -> {2,6}     -> {{1},{1,2}}
%e A371450     65 -> {3,6}     -> {{2},{1,2}}
%e A371450    195 -> {2,3,6}   -> {{1},{2},{1,2}}
%e A371450     11 -> {5}       -> {{3}}
%e A371450     33 -> {2,5}     -> {{1},{3}}
%e A371450     55 -> {3,5}     -> {{2},{3}}
%e A371450    165 -> {2,3,5}   -> {{1},{2},{3}}
%e A371450    143 -> {5,6}     -> {{1,2},{3}}
%e A371450    429 -> {2,5,6}   -> {{1},{1,2},{3}}
%e A371450    715 -> {3,5,6}   -> {{2},{1,2},{3}}
%e A371450   2145 -> {2,3,5,6} -> {{1},{2},{1,2},{3}}
%t A371450 bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A371450 Table[Times@@Prime/@(Times@@Prime/@#&/@bix/@bix[n]),{n,0,30}]
%Y A371450 The sorted version is A329629, with empties A302494.
%Y A371450 A019565 gives Heinz number of binary indices.
%Y A371450 A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
%Y A371450 A070939 gives length of binary expansion.
%Y A371450 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A371450 A326753 counts connected components for BII-numbers, ones A326749.
%Y A371450 Cf. A000720, A003963, A087086, A096111, A275024, A302242, A302505, A302521, A326782, A329557, A329630, A368109.
%K A371450 nonn
%O A371450 0,2
%A A371450 _Gus Wiseman_, Apr 02 2024