A371168 Positive integers with fewer prime factors (A001222) than distinct divisors of prime indices (A370820).

3, 5, 7, 11, 13, 14, 15, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 65, 67, 69, 70, 71, 73, 74, 76, 77, 78, 79, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 105, 106, 107, 109, 111, 113, 114, 115

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:
     3: {2}      35: {3,4}      59: {17}        86: {1,14}
     5: {3}      37: {12}       61: {18}        87: {2,10}
     7: {4}      38: {1,8}      65: {3,6}       89: {24}
    11: {5}      39: {2,6}      67: {19}        91: {4,6}
    13: {6}      41: {13}       69: {2,9}       93: {2,11}
    14: {1,4}    43: {14}       70: {1,3,4}     94: {1,15}
    15: {2,3}    46: {1,9}      71: {20}        95: {3,8}
    17: {7}      47: {15}       73: {21}        97: {25}
    19: {8}      49: {4,4}      74: {1,12}     101: {26}
    21: {2,4}    51: {2,7}      76: {1,1,8}    103: {27}
    23: {9}      52: {1,1,6}    77: {4,5}      105: {2,3,4}
    26: {1,6}    53: {16}       78: {1,2,6}    106: {1,16}
    29: {10}     55: {3,5}      79: {22}       107: {28}
    31: {11}     57: {2,8}      83: {23}       109: {29}
    33: {2,5}    58: {1,10}     85: {3,7}      111: {2,12}
		

Crossrefs

The opposite version is A370348 counted by A371171.

The version for equality is A370802, counted by A371130, strict A371128.

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

For divisors instead of prime factors on the LHS we get A371166.

The complement is counted by A371169.

The weak version is A371170.

Partitions of this type are counted by A371173.

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, A370814, A371127.

Programs

Mathematica

Select[Range[100],PrimeOmega[#]
				

Formula

A001222(a(n)) < A370820(a(n)).