A371172 -id:A371172 - cp's OEIS frontend

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

Gus Wiseman, Mar 19 2024

nonn

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.

			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}

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.

Mathematica
```
Select[Range[100],PrimeNu[#]
				
```

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

A371180 Number of strict integer partitions of n with fewer parts than distinct divisors of parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 2, 4, 4, 7, 8, 10, 12, 15, 19, 22, 29, 33, 40, 47, 57, 68, 81, 95, 110, 129, 152, 178, 207, 240, 277, 317, 365, 422, 486, 558, 632, 723, 824, 940, 1067, 1210, 1371, 1544, 1751, 1977, 2233, 2508, 2820, 3162, 3555, 3983, 4465, 4990, 5571, 6224

Offset: 0

Gus Wiseman, Mar 18 2024

nonn

			The strict partition (6,4,2,1) has 4 parts and 5 distinct divisors of parts {1,2,3,4,5}, so is counted under a(13).
The a(2) = 1 through a(11) = 10 partitions:
  (2)  (3)  (4)  (5)    (6)    (7)    (8)      (9)      (10)     (11)
                 (3,2)  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)    (6,5)
                 (4,1)         (5,2)  (6,2)    (6,3)    (7,3)    (7,4)
                               (6,1)  (4,3,1)  (7,2)    (8,2)    (8,3)
                                               (8,1)    (9,1)    (9,2)
                                               (4,3,2)  (5,3,2)  (10,1)
                                               (6,2,1)  (5,4,1)  (5,4,2)
                                                        (6,3,1)  (6,3,2)
                                                                 (6,4,1)
                                                                 (8,2,1)

The LHS is represented by A001221, distinct case of A001222.
The RHS is represented by A370820, for prime factors A303975.
The version for equality is A371128.
The non-strict version is A371132, ranks A371179.
The non-strict complement is A371178, ranks A371177.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Cf. A003963, A239312, A319055, A355529, A370803, A370808, A370813, A371130 (A370802), A371171, A371172 (A371165), A371173 (A371168).

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[Union[#]] < Length[Union@@Divisors/@#]&]],{n,0,30}]

cp's OEIS Frontend

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

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