|
%I #5 Nov 19 2019 16:35:58
%S 1,2,13,26,29,43,47,58,73,79,86,94,101,113,137,139,146,149,158,163,
%T 167,181,199,202,226,233,257,269,271,274,278,293,298,313,317,326,334,
%U 347,349,362,373,377,389,397,398,421,439,443,449,466,467,487,491,499,514
%N Products of distinct primes of nonprime squarefree index.
%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}}. This sequence lists all MM-numbers of sets of non-singleton sets.
%e The sequence of terms together with their corresponding sets of sets begins:
%e 1: {}
%e 2: {{}}
%e 13: {{1,2}}
%e 26: {{},{1,2}}
%e 29: {{1,3}}
%e 43: {{1,4}}
%e 47: {{2,3}}
%e 58: {{},{1,3}}
%e 73: {{2,4}}
%e 79: {{1,5}}
%e 86: {{},{1,4}}
%e 94: {{},{2,3}}
%e 101: {{1,6}}
%e 113: {{1,2,3}}
%e 137: {{2,5}}
%e 139: {{1,7}}
%e 146: {{},{2,4}}
%e 149: {{3,4}}
%e 158: {{},{1,5}}
%e 163: {{1,8}}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&!MemberQ[primeMS[#],_?PrimeQ]&]
%Y MM-numbers of sets of nonempty sets are A329629.
%Y Products of primes of nonprime squarefree index are A320630.
%Y Products of prime numbers of squarefree index are A302478.
%Y Products of primes of nonprime index are A320628.
%Y Cf. A005117, A056239, A112798, A302242, A302494, A329557.
%Y Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).
%K nonn
%O 1,2
%A _Gus Wiseman_, Nov 18 2019
|
