%I #9 Dec 14 2023 16:29:39

%S 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 65,67,69,71,73,77,79,83,85,87,89,91,93,95,97,101,103,107,109,111,113,

%U 119,123,127,129,131,137,139,141,143,145,149,151,155,157,161,163

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

%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.

%e 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 The terms together with their prime indices of prime indices begin:

%e 1: {}

%e 3: {{1}}

%e 5: {{2}}

%e 7: {{1,1}}

%e 11: {{3}}

%e 13: {{1,2}}

%e 15: {{1},{2}}

%e 17: {{4}}

%e 19: {{1,1,1}}

%e 23: {{2,2}}

%e 29: {{1,3}}

%e 31: {{5}}

%e 33: {{1},{3}}

%e 35: {{2},{1,1}}

%e 37: {{1,1,2}}

%e 39: {{1},{1,2}}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100], Select[Tuples[prix/@prix[#]], UnsameQ@@#&]!={}&]

%Y The complement is A355529, odd A355535, binary A367907.

%Y Positions of positive terms in A367771.

%Y The version for binary indices is A367906, positive positions in A367905.

%Y For a unique choice we have A368101, binary A367908.

%Y The version for divisors instead of factors is A368110, complement A355740.

%Y A058891 counts set-systems, covering A003465, connected A323818.

%Y A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

%Y A124010 gives prime signature, sorted A118914, length A001221, sum A001222.

%Y Cf. A092918, A355737, A355739, A355741, A355744, A355745, A367902.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 12 2023