|
%I #5 Dec 16 2018 17:59:02
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
%T 27,28,29,30,31,32,33,34,36,38,40,41,42,43,44,46,47,48,49,50,51,52,53,
%U 54,55,56,57,58,59,60,62,63,64,66,67,68,72,73,76,79,80
%N Numbers whose product of prime indices is a power of a squarefree number (A072774).
%C The complement is {35, 37, 39, 45, 61, 65, ...}.
%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 multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of regular multiset multisystems, where regularity means all vertex-degrees are equal.
%e Most small numbers belong to this sequence. However, the sequence of multiset multisystems whose MM-numbers do not belong to this sequence begins:
%e 35: {{2},{1,1}}
%e 37: {{1,1,2}}
%e 39: {{1},{1,2}}
%e 45: {{1},{1},{2}}
%e 61: {{1,2,2}}
%e 65: {{2},{1,2}}
%e 69: {{1},{2,2}}
%e 70: {{},{2},{1,1}}
%e 71: {{1,1,3}}
%e 74: {{},{1,1,2}}
%e 75: {{1},{2},{2}}
%e 77: {{1,1},{3}}
%e 78: {{},{1},{1,2}}
%e 87: {{1},{1,3}}
%e 89: {{1,1,1,2}}
%e 90: {{},{1},{1},{2}}
%e 91: {{1,1},{1,2}}
%e 95: {{2},{1,1,1}}
%e 99: {{1},{1},{3}}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]&]
%Y Cf. A003963, A056239, A062503, A072774, A112798, A302242, A302491, A320325, A320698, A322553.
%K nonn
%O 1,2
%A _Gus Wiseman_, Dec 15 2018
|
