A371179 Positive integers with fewer distinct prime factors (A001221) than distinct divisors of prime indices (A370820).

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 63, 65, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101

Offset

1,1

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.

Links

Table of n, a(n) for n=1..66.

Formula

A001221(a(n)) < A370820(a(n)).

Example

The terms together with their prime indices begin:
     3: {2}        28: {1,1,4}    52: {1,1,6}      74: {1,12}
     5: {3}        29: {10}       53: {16}         75: {2,3,3}
     7: {4}        31: {11}       55: {3,5}        76: {1,1,8}
     9: {2,2}      33: {2,5}      56: {1,1,1,4}    77: {4,5}
    11: {5}        35: {3,4}      57: {2,8}        78: {1,2,6}
    13: {6}        37: {12}       58: {1,10}       79: {22}
    14: {1,4}      38: {1,8}      59: {17}         81: {2,2,2,2}
    15: {2,3}      39: {2,6}      61: {18}         83: {23}
    17: {7}        41: {13}       63: {2,2,4}      85: {3,7}
    19: {8}        43: {14}       65: {3,6}        86: {1,14}
    21: {2,4}      45: {2,2,3}    67: {19}         87: {2,10}
    23: {9}        46: {1,9}      69: {2,9}        89: {24}
    25: {3,3}      47: {15}       70: {1,3,4}      91: {4,6}
    26: {1,6}      49: {4,4}      71: {20}         92: {1,1,9}
    27: {2,2,2}    51: {2,7}      73: {21}         93: {2,11}

Mathematica

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

Crossrefs

The LHS is A001221, distinct case of A001222.

The RHS is A370820, for prime factors A303975.

Partitions of this type are counted by A371132, strict A371180.

Counting all prime indices on the LHS gives A371168, counted by A371173.

The complement is A371177, counted by A371178, strict A371128.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length.

A305148 counts pairwise indivisible (stable) partitions, ranks A316476.

Cf. A003963, A239312, A285573, A355529, A370802, A370803, A371130, A371165, A371171, A371172.

Sequence in context: A214547 A097218 A196546 * A231773 A007617 A065878

Adjacent sequences: A371176 A371177 A371178 * A371180 A371181 A371182

Keyword

nonn

Author

Gus Wiseman, Mar 19 2024

Status

approved