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Numbers of which there is exactly one way to choose a different prime factor of each prime index.
11

%I #12 Mar 23 2024 22:13:09

%S 1,3,5,11,15,17,31,33,39,41,51,55,59,65,67,83,85,87,93,109,111,123,

%T 127,129,155,157,165,177,179,187,191,201,205,211,213,235,237,241,249,

%U 255,267,277,283,295,303,305,319,321,327,331,335,341,353,365,367,381

%N Numbers of which there is exactly one way 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 2795 are {3,6,14}, with prime factors {{3},{2,3},{2,7}}, and the only choice with different terms is {3,2,7}, so 2795 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 11: {{3}}

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

%e 17: {{4}}

%e 31: {{5}}

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

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

%e 41: {{6}}

%e 51: {{1},{4}}

%e 55: {{2},{3}}

%e 59: {{7}}

%e 65: {{2},{1,2}}

%e 67: {{8}}

%e 83: {{9}}

%e 85: {{2},{4}}

%e 87: {{1},{1,3}}

%e 93: {{1},{5}}

%e 109: {{10}}

%e 111: {{1},{1,1,2}}

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

%t Select[Range[100], Length[Select[Tuples[prix/@prix[#]], UnsameQ@@#&]]==1&]

%Y For no choices we have A355529, odd A355535, binary A367907.

%Y Positions of ones in A367771.

%Y The version for binary indices is A367908, positions of ones in A367905.

%Y For any number of choices we have A368100.

%Y For a unique set instead of sequence we have A370647, counted by A370594.

%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, sort A118914, length A001221, sum A001222.

%Y A355741 chooses a prime factor of each prime index, multisets A355744.

%Y Cf. A007716, A355737, A355739, A355740, A355745, A367904, A367906, A370584.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 12 2023