cp's OEIS Frontend

A322911 Numbers whose prime indices are all powers of the same squarefree number.

%I A322911 #8 Dec 31 2018 13:18:40
%S A322911 1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,17,18,19,20,21,22,23,24,25,26,27,
%T A322911 28,29,31,32,34,36,38,40,41,42,43,44,46,47,48,49,50,52,53,54,56,57,58,
%U A322911 59,62,63,64,67,68,72,73,76,79,80,81,82,83,84,86,88,92
%N A322911 Numbers whose prime indices are all powers of the same squarefree number.
%C A322911 The complement is {15, 30, 33, 35, 37, 39, 45, ...}. First differs from A318991 at a(33) = 38, A318991(33) = 37.
%C A322911 A multiset multisystem is a finite multiset of finite multisets. 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}}. The dual of a multiset multisystem has, for each vertex, one block consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The sequence lists all MM-numbers of multiset multisystems whose dual is constant, i.e. of the form {x,x,x,...,x} for some multiset x.
%e A322911 The prime indices of 756 are {1,1,2,2,2,4}, which are all powers of 2, so 756 belongs to the sequence.
%e A322911 The prime indices of 841 are {10,10}, which are all powers of 10, so 841 belongs to the sequence.
%e A322911 The prime indices of 2645 are {3,9,9}, which are all powers of 3, so 2645 belongs to the sequence.
%e A322911 The prime indices of 3178 are {1,4,49}, which are all powers of squarefree numbers but not of the same squarefree number, so 3178 does not belong to the sequence.
%e A322911 The prime indices of 30599 are {12,144}, which are all powers of the same number 12, but this number is not squarefree, so 30599 does not belong to the sequence.
%e A322911 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (3,2), (3,2,1), (5,2), (4,3), (6,2), (3,2,2), (7,2), (5,3), (3,2,1,1), (6,3), (5,2,1), (9,2), (4,3,1), (3,3,2), (5,4), (6,2,1), (7,3), (10,2), (3,2,2,1), (6,4), (11,2), (8,3), (5,2,2).
%t A322911 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A322911 powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
%t A322911 sqfker[n_]:=Times@@First/@FactorInteger[n];
%t A322911 Select[Range[100],And[And@@powsqfQ/@primeMS[#],SameQ@@sqfker/@DeleteCases[primeMS[#],1]]&]
%Y A322911 Cf. A000688, A000961, A001597, A005117, A023893, A052410, A056239, A072720, A072774, A302242, A302593, A318400, A322847, A322901, A322912.
%K A322911 nonn
%O A322911 1,2
%A A322911 _Gus Wiseman_, Dec 30 2018