A371169 Positive integers with at least as many prime factors (A001222) as distinct divisors of prime indices (A370820).

1, 2, 4, 6, 8, 9, 10, 12, 16, 18, 20, 22, 24, 25, 27, 28, 30, 32, 34, 36, 40, 42, 44, 45, 48, 50, 54, 56, 60, 62, 63, 64, 66, 68, 72, 75, 80, 81, 82, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 108, 110, 112, 118, 120, 121, 124, 125, 126, 128, 132, 134, 135

Offset: 1

Author

Gus Wiseman, Mar 16 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 terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    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}
    25: {3,3}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    32: {1,1,1,1,1}
    34: {1,7}
    36: {1,1,2,2}
		

Crossrefs

The strict version is A370348 counted by A371171.

The case of equality is A370802, counted by A371130, strict A371128.

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

The complement is A371168, counted by A371173.

The opposite version is A371170.

The version for prime factors instead of divisors on the RHS is A319899.

Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).

A000005 counts divisors.

A001221 counts distinct prime factors.

A027746 lists prime factors, indices A112798, length A001222.

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

Cf. A003963, A355737, A355739, A355741, A370808, A370813, A370814, A371127.

Programs

Mathematica

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