cp's OEIS Frontend

A368729 Numbers whose prime indices are prime or semiprime. MM-numbers of labeled multigraphs with loops and half-loops without isolated (uncovered) nodes.

%I A368729 #8 Jan 08 2024 14:30:33
%S A368729 1,3,5,7,9,11,13,15,17,21,23,25,27,29,31,33,35,39,41,43,45,47,49,51,
%T A368729 55,59,63,65,67,69,73,75,77,79,81,83,85,87,91,93,97,99,101,105,109,
%U A368729 115,117,119,121,123,125,127,129,135,137,139,141,143,145,147,149
%N A368729 Numbers whose prime indices are prime or semiprime. MM-numbers of labeled multigraphs with loops and half-loops without isolated (uncovered) nodes.
%C A368729 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}}.
%e A368729 The terms together with the corresponding multigraphs begin:
%e A368729    1: {}
%e A368729    3: {{1}}
%e A368729    5: {{2}}
%e A368729    7: {{1,1}}
%e A368729    9: {{1},{1}}
%e A368729   11: {{3}}
%e A368729   13: {{1,2}}
%e A368729   15: {{1},{2}}
%e A368729   17: {{4}}
%e A368729   21: {{1},{1,1}}
%e A368729   23: {{2,2}}
%e A368729   25: {{2},{2}}
%e A368729   27: {{1},{1},{1}}
%e A368729   29: {{1,3}}
%e A368729   31: {{5}}
%e A368729   33: {{1},{3}}
%e A368729   35: {{2},{1,1}}
%e A368729   39: {{1},{1,2}}
%e A368729   41: {{6}}
%e A368729   43: {{1,4}}
%e A368729   45: {{1},{1},{2}}
%e A368729   47: {{2,3}}
%e A368729   49: {{1,1},{1,1}}
%t A368729 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A368729 Select[Range[100],OddQ[#]&&Max@@Length/@prix/@prix[#]<=2&]
%Y A368729 In the unlabeled case these multigraphs are counted by A320663.
%Y A368729 These are products of primes indexed by elements of A037143 greater than 1.
%Y A368729 For just primes we have A076610, squarefree A302590.
%Y A368729 For just semiprimes we have A339112, squarefree A340020.
%Y A368729 For just half-loops we have A340019.
%Y A368729 This is the odd case of A368728, complement A368833.
%Y A368729 A000607 counts partitions into primes, with ones allowed A034891.
%Y A368729 A001358 lists semiprimes, squarefree A006881.
%Y A368729 A006450, A106349, A322551, A368732 list selected primes.
%Y A368729 A056239 adds up prime indices, row sums of A112798.
%Y A368729 A101048 counts partitions into semiprimes.
%Y A368729 Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320912, A339113.
%K A368729 nonn
%O A368729 1,2
%A A368729 _Gus Wiseman_, Jan 07 2024