A370802 Positive integers with as many prime factors (A001222) as distinct divisors of prime indices (A370820).

1, 2, 6, 9, 10, 22, 25, 28, 30, 34, 42, 45, 62, 63, 66, 75, 82, 92, 98, 99, 102, 104, 110, 118, 121, 134, 140, 147, 152, 153, 156, 166, 170, 186, 210, 218, 228, 230, 232, 234, 246, 254, 260, 275, 276, 279, 289, 308, 310, 314, 315, 330, 342, 343, 344, 348, 350

Offset: 1

Author

Gus Wiseman, Mar 14 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.

All squarefree terms are even.

Examples

			The prime indices of 1617 are {2,4,4,5}, with distinct divisors {1,2,4,5}, so 1617 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}
   28: {1,1,4}
   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}
   92: {1,1,9}
   98: {1,4,4}
   99: {2,2,5}
  102: {1,2,7}
  104: {1,1,1,6}
		

Crossrefs

For factors instead of divisors on the RHS we have A319899.

A version for binary indices is A367917.

For (greater than) instead of (equal) we have A370348, counted by A371171.

The RHS is A370820, for prime factors instead of divisors A303975.

Partitions of this type are counted by A371130, strict A371128.

For divisors instead of factors on LHS we have A371165, counted by A371172.

For only distinct prime factors on LHS we have A371177, counted by A371178.

Other inequalities: A371166, A371167, A371169, A371170.

A000005 counts divisors.

A001221 counts distinct prime factors.

A027746 lists prime factors, A112798 indices, length A001222.

A239312 counts divisor-choosable partitions, ranks A368110.

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

A370320 counts non-divisor-choosable partitions, ranks A355740.

Cf. A000792, A003963, A355529, A355737, A355739, A355741, A368100, A370808, A370813, A370814, A371127.

Programs

Mathematica

Select[Range[100],PrimeOmega[#]==Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

Formula

A001222(a(n)) = A370820(a(n)).