A355737 -id:A355737 - 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

A370820 Number of positive integers that are a divisor of some prime index of n.

Original entry on oeis.org

0, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 4, 3, 3, 1, 2, 2, 4, 2, 3, 2, 3, 2, 2, 4, 2, 3, 4, 3, 2, 1, 3, 2, 4, 2, 6, 4, 4, 2, 2, 3, 4, 2, 3, 3, 4, 2, 3, 2, 3, 4, 5, 2, 3, 3, 4, 4, 2, 3, 6, 2, 3, 1, 4, 3, 2, 2, 4, 4, 6, 2, 4, 6, 3, 4, 4, 4, 4, 2, 2, 2, 2, 3, 3, 4, 4

Offset: 1

Gus Wiseman, Mar 15 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.

This sequence contains all nonnegative integers. In particular, a(prime(n)!) = n.

			2045 has prime indices {3,80} with combined divisors {1,2,3,4,5,8,10,16,20,40,80}, so a(2045) = 11. In fact, 2045 is the least number with this property.

a(prime(n)) = A000005(n).
Positions of ones are A000079 except for 1.
a(n!) = A000720(n).
a(prime(n)!) = a(prime(A005179(n))) = n.
Counting prime factors instead of divisors gives A303975.
Positions of 2's are A371127.
Position of first appearance of n is A371131(n), sorted version A371181.
RHS of A370802, A371128, A371130, A371165-A371170, A371177, A371178.
A001221 counts distinct prime factors.
A003963 gives product of prime indices.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000792, A007416, A048249, A319899, A355737, A355739, A370348, A370808, A370809.

Table[Length[Union@@Divisors/@PrimePi/@First/@If[n==1,{},FactorInteger[n]]],{n,100}]

a(n) = my(list=List(), f=factor(n)); for (i=1, #f~, fordiv(primepi(f[i,1]), d, listput(list, d))); #Set(list); \\ Michel Marcus, May 02 2024

A370802 Positive integers with as many prime factors (A001222) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

1, 2, 6, 9, 10, 22, 25, 28, 30, 34, 42, 45, 62, 63, 66, 75, 82, 92, 98, 99, 102, 104, 110, 118, 121, 134, 140, 147, 152, 153, 156, 166, 170, 186, 210, 218, 228, 230, 232, 234, 246, 254, 260, 275, 276, 279, 289, 308, 310, 314, 315, 330, 342, 343, 344, 348, 350

Offset: 1

Gus Wiseman, Mar 14 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.

All squarefree terms are even.

			The prime indices of 1617 are {2,4,4,5}, with distinct divisors {1,2,4,5}, so 1617 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}
   28: {1,1,4}
   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}
   92: {1,1,9}
   98: {1,4,4}
   99: {2,2,5}
  102: {1,2,7}
  104: {1,1,1,6}

For factors instead of divisors on the RHS we have A319899.
A version for binary indices is A367917.
For (greater than) instead of (equal) we have A370348, counted by A371171.
The RHS is A370820, for prime factors instead of divisors A303975.
Partitions of this type are counted by A371130, strict A371128.
For divisors instead of factors on LHS we have A371165, counted by A371172.
For only distinct prime factors on LHS we have A371177, counted by A371178.
Other inequalities: A371166, A371167, A371169, A371170.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts divisor-choosable partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A370320 counts non-divisor-choosable partitions, ranks A355740.
Cf. A000792, A003963, A355529, A355737, A355739, A355741, A368100, A370808, A370813, A370814, A371127.

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

A001222(a(n)) = A370820(a(n)).

A355735 Number of ways to choose a divisor of each prime index of n (taken in weakly increasing order) such that the result is weakly increasing.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 3, 3, 1, 2, 3, 4, 2, 5, 2, 3, 2, 3, 4, 4, 3, 4, 3, 2, 1, 3, 2, 4, 3, 6, 4, 7, 2, 2, 5, 4, 2, 4, 3, 4, 2, 6, 3, 3, 4, 5, 4, 3, 3, 7, 4, 2, 3, 6, 2, 7, 1, 6, 3, 2, 2, 5, 4, 6, 3, 4, 6, 4, 4, 4, 7, 4, 2, 5, 2, 2, 5, 3, 4, 7

Offset: 1

Gus Wiseman, Jul 16 2022

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 a(15) = 3 ways are: (1,1), (1,3), (2,3).
The a(18) = 3 ways are: (1,1,1), (1,1,2), (1,2,2).
The a(2) = 1 through a(19) = 4 ways:
  1  1  11  1  11  1  111  11  11  1  111  1  11  11  1111  1  111  1
     2      3  12  2       12  13  5  112  2  12  13        7  112  2
                   4       22              3  14  23           122  4
                                           6                        8

Wikipedia, Cartesian product.

Allowing any choice of divisors gives A355731, firsts A355732.
Choosing a multiset instead of sequence gives A355733, firsts A355734.
Positions of first appearances are A355736.
Choosing only prime divisors gives A355745, variations A355741, A355744.
The reverse version is A355749.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061395 selects the maximum prime index.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A076610, A316524, A335433, A335448, A340827, A340852, A344616, A355737, A355739, A355740, A355742.

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

A355734 Least k such that there are exactly n multisets that can be obtained by choosing a divisor of each prime index of k.

Original entry on oeis.org

1, 3, 7, 13, 21, 35, 39, 89, 133, 105, 91, 195, 351, 285, 247, 333, 273, 481, 455, 555, 623, 801, 791, 741, 1359, 1157, 1281, 1335, 1365, 1443, 1977, 1729, 1967, 1869, 2109, 3185, 2373, 2769, 2639, 4361, 3367, 3653, 3885, 3471, 4613, 5883, 5187, 5551, 6327

Offset: 1

Gus Wiseman, Jul 21 2022

nonn

This is the position of first appearance of n in A355733.

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:
    1: {}
    3: {2}
    7: {4}
   13: {6}
   21: {2,4}
   35: {3,4}
   39: {2,6}
   89: {24}
  133: {4,8}
  105: {2,3,4}
   91: {4,6}
  195: {2,3,6}
  351: {2,2,2,6}
For example, the choices for a(12) = 195 are:
  {1,1,1}  {1,2,2}  {1,3,6}
  {1,1,2}  {1,2,3}  {2,2,3}
  {1,1,3}  {1,2,6}  {2,3,3}
  {1,1,6}  {1,3,3}  {2,3,6}

Counting all choices of divisors gives A355732, firsts of A355731.
Positions of first appearances in A355733.
Choosing weakly increasing divisors gives A355736, firsts of A355735.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A076610, A340852, A344606, A355737, A355739, A355740, A355741, A355744, A355747.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
az=Table[Length[Union[Sort/@Tuples[Divisors/@primeMS[n]]]],{n,1000}];
Table[Position[az,k][[1,1]],{k,mnrm[az]}]

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

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, 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

Gus Wiseman, Mar 16 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 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}

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.

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

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

A355747 Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 58, 116, 320, 772, 2170, 4340, 14112, 28224, 78120, 212004, 612232, 1224464, 3873760, 7747520, 24224608, 64595088, 175452168, 350904336

Offset: 0

Gus Wiseman, Jul 20 2022

			The a(0) = 1 through a(4) = 10 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
           {1,2}  {1,1,2}  {1,1,1,2}
                  {1,1,3}  {1,1,1,3}
                  {1,2,3}  {1,1,1,4}
                           {1,1,2,2}
                           {1,1,2,3}
                           {1,1,2,4}
                           {1,1,3,4}
                           {1,2,2,3}
                           {1,2,3,4}

The sum of the same integers is A000096.
The product of the same integers is A000142, Heinz number A070826.
Counting sequences instead of multisets gives A066843.
The integers themselves are the rows of A131818 (shifted).
For prime indices we have A355733, only prime factors A355744.
For prime factors instead of divisors we have A355746, factors A355537.
A000005 counts divisors.
A000040 lists the prime numbers.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
Cf. A000720, A002110, A006218, A076610, A327486, A355538, A355731, A355737, A355741, A355745.

Table[Length[Union[Sort/@Tuples[Divisors/@Range[n]]]],{n,0,10}]

from sympy import divisors
from itertools import count, islice
def agen():
    s = {tuple()}
    for n in count(1):
        yield len(s)
        s = set(tuple(sorted(t+(d,))) for t in s for d in divisors(n))
print(list(islice(agen(), 16))) # Michael S. Branicky, Aug 03 2022

a(n) = A355733(A070826(n)).

a(p) = 2*a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022

a(15)-a(21) from Michael S. Branicky, Aug 03 2022

a(22)-a(23) from Michael S. Branicky, Aug 08 2022

A368101 Numbers of which there is exactly one way to choose a different prime factor of each prime index.

Original entry on oeis.org

1, 3, 5, 11, 15, 17, 31, 33, 39, 41, 51, 55, 59, 65, 67, 83, 85, 87, 93, 109, 111, 123, 127, 129, 155, 157, 165, 177, 179, 187, 191, 201, 205, 211, 213, 235, 237, 241, 249, 255, 267, 277, 283, 295, 303, 305, 319, 321, 327, 331, 335, 341, 353, 365, 367, 381

Offset: 1

Gus Wiseman, Dec 12 2023

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 2795 are {3,6,14}, with prime factors {{3},{2,3},{2,7}}, and the only choice with different terms is {3,2,7}, so 2795 is in the sequence.
The terms together with their prime indices of prime indices begin:
    1: {}
    3: {{1}}
    5: {{2}}
   11: {{3}}
   15: {{1},{2}}
   17: {{4}}
   31: {{5}}
   33: {{1},{3}}
   39: {{1},{1,2}}
   41: {{6}}
   51: {{1},{4}}
   55: {{2},{3}}
   59: {{7}}
   65: {{2},{1,2}}
   67: {{8}}
   83: {{9}}
   85: {{2},{4}}
   87: {{1},{1,3}}
   93: {{1},{5}}
  109: {{10}}
  111: {{1},{1,1,2}}

For no choices we have A355529, odd A355535, binary A367907.
Positions of ones in A367771.
The version for binary indices is A367908, positions of ones in A367905.
For any number of choices we have A368100.
For a unique set instead of sequence we have A370647, counted by A370594.
A058891 counts set-systems, covering A003465, connected A323818.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sort A118914, length A001221, sum A001222.
A355741 chooses a prime factor of each prime index, multisets A355744.
Cf. A007716,  A355737, A355739, A355740, A355745, A367904, A367906, A370584.

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

A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 5, 0, 2, 3, 7, 0, 11, 5, 6, 0, 15, 2, 22, 0, 10, 7, 30, 0, 6, 11, 0, 0, 42, 6, 56, 0, 14, 15, 15, 0, 77, 22, 22, 0, 101, 10, 135, 0, 6, 30, 176, 0, 20, 6, 30, 0, 231, 0, 21, 0, 44, 42, 297, 0, 385, 56, 10, 0, 33, 14, 490, 0, 60, 15, 627, 0

Offset: 1

Gus Wiseman, Aug 18 2025

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 axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			The prime indices of 9 are (2,2), and there are a(9) = 2 choices:
  ((2),(1,1))
  ((1,1),(2))
The prime indices of 15 are (2,3), and there are a(15) = 5 choices:
  ((2),(3))
  ((2),(2,1))
  ((2),(1,1,1))
  ((1,1),(2,1))
  ((1,1),(1,1,1))

Positions of zeros are A276078 (choosable), complement A276079 (non-choosable).
Allowing repeated partitions gives A299200, A357977, A357982, A357978.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors see A355731, A355733, A355734, A355735, A355737, A355739, A355740.
For prime factors instead of partitions see A355741, A355742, A387136.
The disjoint case is A383706.
For initial intervals instead of partitions we have A387111.
The case of strict partitions is A387115.
The case of constant partitions is A387120.
Taking each prime factor (instead of index) gives A387133.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A008284, A063834, A261049, A296122, A299201, A335433, A335448, A355745, A387135.

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

A387111 Number of ways to choose a sequence of distinct positive integers, one in the initial interval of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 1, 4, 0, 2, 2, 5, 0, 6, 3, 4, 0, 7, 0, 8, 0, 6, 4, 9, 0, 6, 5, 0, 0, 10, 1, 11, 0, 8, 6, 9, 0, 12, 7, 10, 0, 13, 2, 14, 0, 2, 8, 15, 0, 12, 2, 12, 0, 16, 0, 12, 0, 14, 9, 17, 0, 18, 10, 4, 0, 15, 3, 19, 0, 16, 4, 20, 0, 21, 11, 4, 0, 16, 4, 22

Offset: 1

Gus Wiseman, Aug 18 2025

nonn

The initial interval of a nonnegative integer x is the set {1,...,x}.
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 axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			The prime indices of 75 are (2,3,3), with initial intervals ({1,2},{1,2,3},{1,2,3}), with choices (1,2,3), (1,3,2), (2,1,3), (2,3,1), so a(75) = 4.

Allowing repeated partitions gives A003963.
For constant instead of distinct we have A055396.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors we have A355739, zeros A355740, strict case of A355731.
For prime factors we have A355741, prime powers A355742, weakly increasing A355745.
For integer partitions we have A387110.
Positions of nonzero terms are A387112 (choosable).
Positions of 0 are A387134 (non-choosable).
A001414 adds up distinct prime divisors, counted by A001221.
A061395 gives greatest prime index.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A326841, A335433, A335448, A355733, A355737, A355747, A357980, A383706, A387120, A387133, A387135.

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

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A370820 Number of positive integers that are a divisor of some prime index of n.

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PARI

A370802 Positive integers with as many prime factors (A001222) as distinct divisors of prime indices (A370820).

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A355735 Number of ways to choose a divisor of each prime index of n (taken in weakly increasing order) such that the result is weakly increasing.

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A355734 Least k such that there are exactly n multisets that can be obtained by choosing a divisor of each prime index of k.

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

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A355747 Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.

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A368101 Numbers of which there is exactly one way to choose a different prime factor of each prime index.

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A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.