A370647 Numbers such that only one set can be obtained by choosing a different prime factor of each prime index.

1, 3, 5, 7, 11, 15, 17, 19, 23, 31, 33, 35, 39, 41, 51, 53, 55, 59, 65, 67, 69, 77, 83, 85, 87, 91, 93, 95, 97, 103, 109, 111, 119, 123, 127, 129, 131, 155, 157, 161, 165, 169, 177, 179, 183, 185, 187, 191, 201, 203, 205, 209, 211, 213, 217, 227, 235, 237, 241

Offset: 1

Author

Gus Wiseman, Mar 06 2024

Keywords

• nonn

Comments

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.

Examples

			The prime indices of 91 are {4,6}, with only choice {2,3}, so 91 is in the sequence.
The terms together with their prime indices begin:
     1: {}        53: {16}      109: {29}
     3: {2}       55: {3,5}     111: {2,12}
     5: {3}       59: {17}      119: {4,7}
     7: {4}       65: {3,6}     123: {2,13}
    11: {5}       67: {19}      127: {31}
    15: {2,3}     69: {2,9}     129: {2,14}
    17: {7}       77: {4,5}     131: {32}
    19: {8}       83: {23}      155: {3,11}
    23: {9}       85: {3,7}     157: {37}
    31: {11}      87: {2,10}    161: {4,9}
    33: {2,5}     91: {4,6}     165: {2,3,5}
    35: {3,4}     93: {2,11}    169: {6,6}
    39: {2,6}     95: {3,8}     177: {2,17}
    41: {13}      97: {25}      179: {41}
    51: {2,7}    103: {27}      183: {2,18}
		

Crossrefs

For nonexistence we have A355529, count A370593.

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

For existence we have A368100, count A370592.

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

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

Maximal sets of this type are counted by A370585.

Partitions of this type are counted by A370594.

For subsets and binary indices we have A370638.

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

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

A006530 gives greatest prime factor, least A020639.

A027746 lists prime factors, A112798 indices, length A001222.

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

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

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Union[Sort /@ Select[Tuples[prix/@prix[#]],UnsameQ@@#&]]]==1&]