A370810 Numbers n such that only one set can be obtained by choosing a different divisor of each prime index of n.

1, 2, 6, 9, 10, 22, 25, 30, 34, 42, 45, 62, 63, 66, 75, 82, 98, 99, 102, 110, 118, 121, 134, 147, 153, 166, 170, 186, 210, 218, 230, 246, 254, 275, 279, 289, 310, 314, 315, 330, 343, 354, 358, 363, 369, 374, 382, 390, 402, 410, 422, 425, 462, 482, 490, 495

Offset: 1

Author

Gus Wiseman, Mar 05 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 6591 are {2,6,6,6}, for which the only choice is {1,2,3,6}, so 6591 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   22: {1,5}
   25: {3,3}
   30: {1,2,3}
   34: {1,7}
   42: {1,2,4}
   45: {2,2,3}
   62: {1,11}
   63: {2,2,4}
   66: {1,2,5}
   75: {2,3,3}
   82: {1,13}
   98: {1,4,4}
   99: {2,2,5}
  102: {1,2,7}
  110: {1,3,5}
		

Crossrefs

For no choices we have A355740, counted by A370320.

For at least one choice we have A368110, counted by A239312.

Partitions of this type are counted by A370595 and A370815.

For just prime factors we have A370647, counted by A370594.

For more than one choice we have A370811, counted by A370803.

A000005 counts divisors.

A006530 gives greatest prime factor, least A020639.

A027746 lists prime factors, A112798 indices, length A001222.

A355731 counts choices of a divisor of each prime index, firsts A355732.

A355741, A355744, A355745 choose prime factors of prime indices.

A370814 counts factorizations with choosable divisors, complement A370813.

Cf. A133686, A355529, A355739, A355749, A367771, A367904, A370584, A370592, A370808, A370816.

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[Divisors/@prix[#]],UnsameQ@@#&]]]==1&]