A371170 Positive integers with at most as many prime factors (A001222) as distinct divisors of prime indices (A370820).

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 92

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 prime indices of 105 are {2,3,4}, and there are 3 prime factors (3,5,7) and 4 distinct divisors of prime indices (1,2,3,4), so 105 is in the sequence.
The terms together with their prime indices begin:
     1: {}       22: {1,5}      42: {1,2,4}    63: {2,2,4}
     2: {1}      23: {9}        43: {14}       65: {3,6}
     3: {2}      25: {3,3}      45: {2,2,3}    66: {1,2,5}
     5: {3}      26: {1,6}      46: {1,9}      67: {19}
     6: {1,2}    28: {1,1,4}    47: {15}       69: {2,9}
     7: {4}      29: {10}       49: {4,4}      70: {1,3,4}
     9: {2,2}    30: {1,2,3}    51: {2,7}      71: {20}
    10: {1,3}    31: {11}       52: {1,1,6}    73: {21}
    11: {5}      33: {2,5}      53: {16}       74: {1,12}
    13: {6}      34: {1,7}      55: {3,5}      75: {2,3,3}
    14: {1,4}    35: {3,4}      57: {2,8}      76: {1,1,8}
    15: {2,3}    37: {12}       58: {1,10}     77: {4,5}
    17: {7}      38: {1,8}      59: {17}       78: {1,2,6}
    19: {8}      39: {2,6}      61: {18}       79: {22}
    21: {2,4}    41: {13}       62: {1,11}     82: {1,13}
		

Crossrefs

The complement 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 strict version is A371168 counted by A371173.

The opposite version is A371169.

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, A319899, A355737, A355739, A355741, A370808, A371127.

Programs

Mathematica

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