A371177 Positive integers whose prime indices include all distinct divisors of all prime indices.

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 30, 32, 34, 36, 40, 42, 44, 48, 50, 54, 60, 62, 64, 66, 68, 72, 80, 82, 84, 88, 90, 96, 100, 102, 108, 110, 118, 120, 124, 126, 128, 132, 134, 136, 144, 150, 160, 162, 164, 166, 168, 170, 176, 180, 186, 192, 198, 200

Offset: 1

Author

Gus Wiseman, Mar 18 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.

Also positive integers with as many distinct prime factors (A001221) as distinct divisors of prime indices (A370820).

Examples

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}
		

Crossrefs

The LHS is A001221, distinct case of A001222.

The RHS is A370820, for prime factors A303975.

For bigomega on the LHS we have A370802, counted by A371130.

For divisors on the LHS we have A371165, counted by A371172.

Partitions of this type are counted by A371178, strict A371128.

The complement is A371179, counted by A371132.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length.

A305148 counts partitions without divisors, strict A303362, ranks A316476.

Cf. A000837, A003963, A239312, A285573, A355529, A370813, A371168.

Programs

Mathematica

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

Formula

A001221(a(n)) = A370820(a(n)).