A319055 -id:A319055 - cp's OEIS frontend

A370817 Greatest number of multisets that can be obtained by choosing a prime factor of each factor in an integer factorization of n into unordered factors > 1.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 07 2024

nonn

First differs from A096825 at a(210) = 4, A096825(210) = 6.
First differs from A343943 at a(210) = 4, A343943(210) = 6.
First differs from A345926 at a(90) = 4, A345926(90) = 3.

			For the factorizations of 60 we have the following choices (using prime indices {1,2,3} instead of prime factors {2,3,5}):
  (2*2*3*5): {{1,1,2,3}}
   (2*2*15): {{1,1,2},{1,1,3}}
   (2*3*10): {{1,1,2},{1,2,3}}
    (2*5*6): {{1,1,3},{1,2,3}}
    (3*4*5): {{1,2,3}}
     (2*30): {{1,1},{1,2},{1,3}}
     (3*20): {{1,2},{2,3}}
     (4*15): {{1,2},{1,3}}
     (5*12): {{1,3},{2,3}}
     (6*10): {{1,1},{1,2},{1,3},{2,3}}
       (60): {{1},{2},{3}}
So a(60) = 4.

Max Alekseyev, Table of n, a(n) for n = 1..10000

For all divisors (not just prime factors) we have A370816.
The version for partitions is A370809, for all divisors A370808.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
A368413 counts non-choosable factorizations, complement A368414.
A370813 counts non-divisor-choosable factorizations, complement A370814.
Cf. A000792, A048249, A066739, A319055, A319057, A355529, A355535, A367771, A368100, A370585.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Max[Length[Union[Sort/@Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#]]]&/@facs[n]],{n,100}]

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

A370817 Greatest number of multisets that can be obtained by choosing a prime factor of each factor in an integer factorization of n into unordered factors > 1.

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