A370647 - OEIS

%I #8 Mar 06 2024 14:47:42

%S 1,3,5,7,11,15,17,19,23,31,33,35,39,41,51,53,55,59,65,67,69,77,83,85,

%T 87,91,93,95,97,103,109,111,119,123,127,129,131,155,157,161,165,169,

%U 177,179,183,185,187,191,201,203,205,209,211,213,217,227,235,237,241

%N Numbers such that only one set can be obtained by choosing 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 91 are {4,6}, with only choice {2,3}, so 91 is in the sequence.

%e The terms together with their prime indices begin:

%e      1: {}        53: {16}      109: {29}

%e      3: {2}       55: {3,5}     111: {2,12}

%e      5: {3}       59: {17}      119: {4,7}

%e      7: {4}       65: {3,6}     123: {2,13}

%e     11: {5}       67: {19}      127: {31}

%e     15: {2,3}     69: {2,9}     129: {2,14}

%e     17: {7}       77: {4,5}     131: {32}

%e     19: {8}       83: {23}      155: {3,11}

%e     23: {9}       85: {3,7}     157: {37}

%e     31: {11}      87: {2,10}    161: {4,9}

%e     33: {2,5}     91: {4,6}     165: {2,3,5}

%e     35: {3,4}     93: {2,11}    169: {6,6}

%e     39: {2,6}     95: {3,8}     177: {2,17}

%e     41: {13}      97: {25}      179: {41}

%e     51: {2,7}    103: {27}      183: {2,18}

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

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

%Y For nonexistence we have A355529, count A370593.

%Y For binary instead of prime indices we have A367908, counted by A367904.

%Y For existence we have A368100, count A370592.

%Y For a sequence instead of set of factors we have A368101.

%Y The version for subsets is A370584, see also A370582, A370583.

%Y Maximal sets of this type are counted by A370585.

%Y Partitions of this type are counted by A370594.

%Y For subsets and binary indices we have A370638.

%Y The version for factorizations is A370645, see also A368414, A368413.

%Y For divisors instead of factors we have A370810, counted by A370595.

%Y A006530 gives greatest prime factor, least A020639.

%Y A027746 lists prime factors, A112798 indices, length A001222.

%Y A355741 counts ways to choose a prime factor of each prime index.

%Y Cf. A000040, A000720, A003963, A355739, A355740, A355744, A355745, A368110.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 06 2024