A355535 Odd numbers of which it is not possible to choose a different prime factor of each prime index.

9, 21, 25, 27, 45, 49, 57, 63, 75, 81, 99, 105, 115, 117, 121, 125, 133, 135, 147, 153, 159, 171, 175, 189, 195, 207, 225, 231, 243, 245, 261, 273, 275, 279, 285, 289, 297, 315, 325, 333, 343, 345, 351, 357, 361, 363, 369, 371, 375, 387, 393, 399, 405, 423

Offset: 1

Author

Gus Wiseman, Jul 22 2022

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 terms together with their prime indices begin:
    9: {2,2}
   21: {2,4}
   25: {3,3}
   27: {2,2,2}
   45: {2,2,3}
   49: {4,4}
   57: {2,8}
   63: {2,2,4}
   75: {2,3,3}
   81: {2,2,2,2}
   99: {2,2,5}
  105: {2,3,4}
For example, the prime indices of 897 are {2,6,9}, of which we can choose prime factors in two ways: (2,2,3) or (2,3,3); but neither of these has all distinct elements, so 897 is in the sequence.
		

Crossrefs

Including evens gives A355529.

The version for all divisors including evens is A355740, zeros of A355739.

Choices of a prime factor of each prime index: A355741, unordered A355744.

A001221 counts distinct prime factors, with sum A001414.

A001222 counts prime factors with multiplicity.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798.

A120383 lists numbers divisible by all of their prime indices.

Cf. A000720, A076610, A289509, A302796, A327486, A355731, A355733, A355742.

Programs

Mathematica

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