A304636 Numbers n with prime omicron 3, meaning A304465(n) = 3.

8, 27, 30, 42, 66, 70, 78, 102, 105, 110, 114, 125, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 343, 345, 354, 357, 360, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426

Offset: 1

Author

Gus Wiseman, May 15 2018

Keywords

• nonn

Comments

If n > 1 is not a prime number, we have A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= omicron(n) > 1 where Omega = A001222, omega = A001221, and omicron = A304465.

Examples

			This is a list of normalized factorizations (see A112798) of selected entries:
     8: {1,1,1}
    30: {1,2,3}
   360: {1,1,1,2,2,3}
   720: {1,1,1,1,2,2,3}
   900: {1,1,2,2,3,3}
  1440: {1,1,1,1,1,2,2,3}
  2160: {1,1,1,1,2,2,2,3}
  2880: {1,1,1,1,1,1,2,2,3}
  4320: {1,1,1,1,1,2,2,2,3}
  5760: {1,1,1,1,1,1,1,2,2,3}
  8640: {1,1,1,1,1,1,2,2,2,3}
Starting with A112798(1801800) and repeatedly taking the multiset of multiplicities we have {1,1,1,2,2,3,3,4,5,6} -> {1,1,1,2,2,3} -> {1,2,3} -> {1,1,1} -> {3}, so 1801800 belongs to the sequence.
		

Crossrefs

Cf. A001221, A001222, A005117, A007916, A014612, A033992, A071625, A112798, A181819, A182850, A182857, A304464, A304465, A304634, A304647.

Programs

Mathematica

Join@@Position[Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,200}],3]