OFFSET
0,2
COMMENTS
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}}.
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.
EXAMPLE
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.
The terms together with their prime indices and binary indices of prime indices begin:
1 -> {} -> {}
3 -> {2} -> {{1}}
5 -> {3} -> {{2}}
15 -> {2,3} -> {{1},{2}}
13 -> {6} -> {{1,2}}
39 -> {2,6} -> {{1},{1,2}}
65 -> {3,6} -> {{2},{1,2}}
195 -> {2,3,6} -> {{1},{2},{1,2}}
11 -> {5} -> {{3}}
33 -> {2,5} -> {{1},{3}}
55 -> {3,5} -> {{2},{3}}
165 -> {2,3,5} -> {{1},{2},{3}}
143 -> {5,6} -> {{1,2},{3}}
429 -> {2,5,6} -> {{1},{1,2},{3}}
715 -> {3,5,6} -> {{2},{1,2},{3}}
2145 -> {2,3,5,6} -> {{1},{2},{1,2},{3}}
MATHEMATICA
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Times@@Prime/@(Times@@Prime/@#&/@bix/@bix[n]), {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 02 2024
STATUS
approved