A370816 -id:A370816 - cp's OEIS frontend

A370808 Greatest number of multisets that can be obtained by choosing a divisor of each part of an integer partition of n.

1, 1, 2, 2, 3, 4, 5, 6, 7, 10, 11, 14, 17, 19, 23, 29, 30, 39, 41, 51, 58, 66, 78, 82, 102, 110, 132, 144, 162, 186, 210, 228, 260, 296, 328, 366, 412, 462, 512, 560, 638, 692, 764, 860, 924, 1028, 1122, 1276, 1406, 1528, 1721, 1898, 2056, 2318, 2506, 2812, 3020, 3442

Offset: 0

Gus Wiseman, Mar 05 2024

nonn

			For the partitions of 5 we have the following choices:
      (5): {{1},{5}}
     (41): {{1,1},{1,2},{1,4}}
     (32): {{1,1},{1,2},{1,3},{2,3}}
    (311): {{1,1,1},{1,1,3}}
    (221): {{1,1,1},{1,1,2},{1,2,2}}
   (2111): {{1,1,1,1},{1,1,1,2}}
  (11111): {{1,1,1,1,1}}
So a(5) = 4.

For just prime factors we have A370809.
The version for factorizations is A370816, for just prime factors A370817.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts condensed partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355733 counts choices of divisors of prime indicec.
A370320 counts non-condensed partitions, ranks A355740.
A370592 counts factor-choosable partitions, complement A370593.
Cf. A000792, A048249, A066739, A319055, A355737, A355739, A370348, A370595, A370803, A370810.

Table[Max[Length[Union[Sort/@Tuples[Divisors/@#]]]&/@IntegerPartitions[n]],{n,0,30}]

Terms a(31) onward from Max Alekseyev, Sep 17 2024

A370810 Numbers n such that only one set can be obtained by choosing a different divisor of each prime index of n.

Original entry on oeis.org

1, 2, 6, 9, 10, 22, 25, 30, 34, 42, 45, 62, 63, 66, 75, 82, 98, 99, 102, 110, 118, 121, 134, 147, 153, 166, 170, 186, 210, 218, 230, 246, 254, 275, 279, 289, 310, 314, 315, 330, 343, 354, 358, 363, 369, 374, 382, 390, 402, 410, 422, 425, 462, 482, 490, 495

Offset: 1

Gus Wiseman, Mar 05 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 prime indices of 6591 are {2,6,6,6}, for which the only choice is {1,2,3,6}, so 6591 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   22: {1,5}
   25: {3,3}
   30: {1,2,3}
   34: {1,7}
   42: {1,2,4}
   45: {2,2,3}
   62: {1,11}
   63: {2,2,4}
   66: {1,2,5}
   75: {2,3,3}
   82: {1,13}
   98: {1,4,4}
   99: {2,2,5}
  102: {1,2,7}
  110: {1,3,5}

For no choices we have A355740, counted by A370320.
For at least one choice we have A368110, counted by A239312.
Partitions of this type are counted by A370595 and A370815.
For just prime factors we have A370647, counted by A370594.
For more than one choice we have A370811, counted by A370803.
A000005 counts divisors.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741, A355744, A355745 choose prime factors of prime indices.
A370814 counts factorizations with choosable divisors, complement A370813.
Cf. A133686, A355529, A355739, A355749, A367771, A367904, A370584, A370592, A370808, A370816.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Union[Sort /@ Select[Tuples[Divisors/@prix[#]],UnsameQ@@#&]]]==1&]

A370809 Greatest number of multisets that can be obtained by choosing a prime factor of each part of an integer partition of n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 4, 4, 6, 4, 6, 6, 6, 6, 8, 6, 8, 8, 9, 8, 10, 9, 12, 10, 12, 12, 12, 12, 16, 13, 16, 16, 18, 16, 20, 18, 20, 20, 24, 20, 24, 24, 24, 26, 30, 26, 30, 30, 32, 32, 36, 32, 36, 36, 40, 38, 42, 40, 45, 44, 48

Offset: 0

Gus Wiseman, Mar 05 2024

nonn

			For the partition (10,6,3,2) there are 4 choices: {2,2,2,3}, {2,2,3,3}, {2,2,3,5}, {2,3,3,5} so a(21) >= 4.
For the partitions of 6 we have the following choices:
  (6): {{2},{3}}
  (51): {}
  (42): {{2,2}}
  (411): {}
  (33): {{3,3}}
  (321): {}
  (3111): {}
  (222): {{2,2,2}}
  (2211): {}
  (21111): {}
  (111111): {}
So a(6) = 2.

For just all divisors (not just prime factors) we have A370808.
The version for factorizations is A370817, for all divisors A370816.
A000041 counts integer partitions, strict A000009.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741, A355744, A355745 choose prime factors of prime indices.
A368413 counts non-choosable factorizations, complement A368414.
A370320 counts non-condensed partitions, ranks A355740.
A370592, A370593, A370594, `A370807 count non-choosable partitions.
Cf. A000792, A048249, A063834, A239312, A319055, A339095, A355529, A355733, A367771, A368100, A370585.

Table[Max[Length[Union[Sort /@ Tuples[If[#==1,{},First/@FactorInteger[#]]& /@ #]]]&/@IntegerPartitions[n]],{n,0,30}]

Terms a(31) onward from Max Alekseyev, Sep 17 2024

A370811 Numbers such that more than one set can be obtained by choosing a different divisor of each prime index.

Original entry on oeis.org

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, 53, 55, 57, 58, 59, 61, 65, 67, 69, 70, 71, 73, 74, 77, 78, 79, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 105, 106, 107, 109, 111, 113, 114, 115, 117, 119

Offset: 1

Gus Wiseman, Mar 13 2024

nonn

A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798.

			The prime indices of 70 are {1,3,4}, with choices (1,3,4) and (1,3,2), so 70 is in the sequence.
The terms together with their prime indices begin:
     3: {2}      43: {14}        79: {22}       115: {3,9}
     5: {3}      46: {1,9}       83: {23}       117: {2,2,6}
     7: {4}      47: {15}        85: {3,7}      119: {4,7}
    11: {5}      49: {4,4}       86: {1,14}     122: {1,18}
    13: {6}      51: {2,7}       87: {2,10}     123: {2,13}
    14: {1,4}    53: {16}        89: {24}       127: {31}
    15: {2,3}    55: {3,5}       91: {4,6}      129: {2,14}
    17: {7}      57: {2,8}       93: {2,11}     130: {1,3,6}
    19: {8}      58: {1,10}      94: {1,15}     131: {32}
    21: {2,4}    59: {17}        95: {3,8}      133: {4,8}
    23: {9}      61: {18}        97: {25}       137: {33}
    26: {1,6}    65: {3,6}      101: {26}       138: {1,2,9}
    29: {10}     67: {19}       103: {27}       139: {34}
    31: {11}     69: {2,9}      105: {2,3,4}    141: {2,15}
    33: {2,5}    70: {1,3,4}    106: {1,16}     142: {1,20}
    35: {3,4}    71: {20}       107: {28}       143: {5,6}
    37: {12}     73: {21}       109: {29}       145: {3,10}
    38: {1,8}    74: {1,12}     111: {2,12}     146: {1,21}
    39: {2,6}    77: {4,5}      113: {30}       149: {35}
    41: {13}     78: {1,2,6}    114: {1,2,8}    151: {36}

For no choices we have A355740, counted by A370320.
For at least one choice we have A368110, counted by A239312.
Partitions of this type are counted by A370803.
For a unique choice we have A370810, counted by A370595 and A370815.
A000005 counts divisors.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741, A355744, A355745 choose prime factors of prime indices.
A370814 counts factorizations with choosable divisors, complement A370813.
Cf. A133686, A355529, A355739, A355749, A367771, A367904, A370584, A370592, A370594, A370647, A370808, A370816.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Union[Sort /@ Select[Tuples[Divisors/@prix[#]],UnsameQ@@#&]]]>1&]

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.

Original entry on oeis.org

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}]

cp's OEIS Frontend

A370808 Greatest number of multisets that can be obtained by choosing a divisor of each part of an integer partition of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A370810 Numbers n such that only one set can be obtained by choosing a different divisor of each prime index of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A370809 Greatest number of multisets that can be obtained by choosing a prime factor of each part of an integer partition of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A370811 Numbers such that more than one set can be obtained by choosing a different divisor of each prime index.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica