cp's OEIS Frontend

A368100 Numbers of which it is possible to choose a different prime factor of each prime index.

%I A368100 #9 Dec 14 2023 16:29:39
%S A368100 1,3,5,7,11,13,15,17,19,23,29,31,33,35,37,39,41,43,47,51,53,55,59,61,
%T A368100 65,67,69,71,73,77,79,83,85,87,89,91,93,95,97,101,103,107,109,111,113,
%U A368100 119,123,127,129,131,137,139,141,143,145,149,151,155,157,161,163
%N A368100 Numbers of which it is possible to choose a different prime factor of each prime index.
%C A368100 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.
%e A368100 The prime indices of 2849 are {4,5,12}, with prime factors {{2,2},{5},{2,2,3}}, and of the two choices (2,5,2) and (2,5,3) the latter has all different terms, so 2849 is in the sequence.
%e A368100 The terms together with their prime indices of prime indices begin:
%e A368100    1: {}
%e A368100    3: {{1}}
%e A368100    5: {{2}}
%e A368100    7: {{1,1}}
%e A368100   11: {{3}}
%e A368100   13: {{1,2}}
%e A368100   15: {{1},{2}}
%e A368100   17: {{4}}
%e A368100   19: {{1,1,1}}
%e A368100   23: {{2,2}}
%e A368100   29: {{1,3}}
%e A368100   31: {{5}}
%e A368100   33: {{1},{3}}
%e A368100   35: {{2},{1,1}}
%e A368100   37: {{1,1,2}}
%e A368100   39: {{1},{1,2}}
%t A368100 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A368100 Select[Range[100], Select[Tuples[prix/@prix[#]], UnsameQ@@#&]!={}&]
%Y A368100 The complement is A355529, odd A355535, binary A367907.
%Y A368100 Positions of positive terms in A367771.
%Y A368100 The version for binary indices is A367906, positive positions in A367905.
%Y A368100 For a unique choice we have A368101, binary A367908.
%Y A368100 The version for divisors instead of factors is A368110, complement A355740.
%Y A368100 A058891 counts set-systems, covering A003465, connected A323818.
%Y A368100 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A368100 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A368100 Cf. A092918,  A355737, A355739, A355741, A355744, A355745, A367902.
%K A368100 nonn
%O A368100 1,2
%A A368100 _Gus Wiseman_, Dec 12 2023