cp's OEIS Frontend

A368101 Numbers of which there is exactly one way to choose a different prime factor of each prime index.

%I A368101 #12 Mar 23 2024 22:13:09
%S A368101 1,3,5,11,15,17,31,33,39,41,51,55,59,65,67,83,85,87,93,109,111,123,
%T A368101 127,129,155,157,165,177,179,187,191,201,205,211,213,235,237,241,249,
%U A368101 255,267,277,283,295,303,305,319,321,327,331,335,341,353,365,367,381
%N A368101 Numbers of which there is exactly one way to choose a different prime factor of each prime index.
%C A368101 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 A368101 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 A368101 The terms together with their prime indices of prime indices begin:
%e A368101     1: {}
%e A368101     3: {{1}}
%e A368101     5: {{2}}
%e A368101    11: {{3}}
%e A368101    15: {{1},{2}}
%e A368101    17: {{4}}
%e A368101    31: {{5}}
%e A368101    33: {{1},{3}}
%e A368101    39: {{1},{1,2}}
%e A368101    41: {{6}}
%e A368101    51: {{1},{4}}
%e A368101    55: {{2},{3}}
%e A368101    59: {{7}}
%e A368101    65: {{2},{1,2}}
%e A368101    67: {{8}}
%e A368101    83: {{9}}
%e A368101    85: {{2},{4}}
%e A368101    87: {{1},{1,3}}
%e A368101    93: {{1},{5}}
%e A368101   109: {{10}}
%e A368101   111: {{1},{1,1,2}}
%t A368101 prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A368101 Select[Range[100], Length[Select[Tuples[prix/@prix[#]], UnsameQ@@#&]]==1&]
%Y A368101 For no choices we have A355529, odd A355535, binary A367907.
%Y A368101 Positions of ones in A367771.
%Y A368101 The version for binary indices is A367908, positions of ones in A367905.
%Y A368101 For any number of choices we have A368100.
%Y A368101 For a unique set instead of sequence we have A370647, counted by A370594.
%Y A368101 A058891 counts set-systems, covering A003465, connected A323818.
%Y A368101 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A368101 A124010 gives prime signature, sort A118914, length A001221, sum A001222.
%Y A368101 A355741 chooses a prime factor of each prime index, multisets A355744.
%Y A368101 Cf. A007716,  A355737, A355739, A355740, A355745, A367904, A367906, A370584.
%K A368101 nonn
%O A368101 1,2
%A A368101 _Gus Wiseman_, Dec 12 2023