cp's OEIS Frontend

A319899 Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963).

%I A319899 #26 Dec 18 2018 17:07:29
%S A319899 1,3,5,7,11,15,17,19,23,26,31,33,35,39,41,51,53,55,58,59,65,67,69,74,
%T A319899 77,83,85,86,87,91,93,94,95,97,103,109,111,119,122,123,127,129,131,
%U A319899 142,146,155,157,158,161,165,169,177,178,179,183,185,187,191,201,202
%N A319899 Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963).
%C A319899 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 square multiset multisystems, meaning the number of edges is equal to the number of distinct vertices.
%e A319899 The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:
%e A319899    1: {}
%e A319899    3: {{1}}
%e A319899    5: {{2}}
%e A319899    7: {{1,1}}
%e A319899   11: {{3}}
%e A319899   15: {{1},{2}}
%e A319899   17: {{4}}
%e A319899   19: {{1,1,1}}
%e A319899   23: {{2,2}}
%e A319899   26: {{},{1,2}}
%e A319899   31: {{5}}
%e A319899   33: {{1},{3}}
%e A319899   35: {{2},{1,1}}
%e A319899   39: {{1},{1,2}}
%e A319899   41: {{6}}
%e A319899   51: {{1},{4}}
%e A319899   53: {{1,1,1,1}}
%e A319899   55: {{2},{3}}
%e A319899   58: {{},{1,3}}
%e A319899   59: {{7}}
%e A319899   65: {{2},{1,2}}
%e A319899   67: {{8}}
%e A319899   69: {{1},{2,2}}
%e A319899   74: {{},{1,1,2}}
%t A319899 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A319899 Select[Range[100],PrimeOmega[#]==PrimeNu[Times@@primeMS[#]]&]
%Y A319899 Cf. A003963, A057151, A064573, A120732, A319616, A319877, A320325, A322527, A322530.
%K A319899 nonn
%O A319899 1,2
%A A319899 _Gus Wiseman_, Dec 17 2018