cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A370595 Number of integer partitions of n such that only one set can be obtained by choosing a different divisor of each part.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 3, 2, 4, 3, 4, 5, 8, 9, 8, 13, 12, 17, 16, 27, 28, 33, 36, 39, 50, 58, 65, 75, 93, 94, 112, 125, 148, 170, 190, 209, 250, 273, 305, 341, 403, 432, 484, 561, 623, 708, 765, 873, 977, 1109, 1178, 1367, 1493, 1669, 1824, 2054, 2265, 2521, 2770
Offset: 0

Views

Author

Gus Wiseman, Mar 03 2024

Keywords

Comments

For example, the only choice for the partition (9,9,6,6,6) is {1,2,3,6,9}.

Examples

			The a(1) = 1 through a(15) = 13 partitions (A = 10, B = 11, C = 12, D = 13):
  1  .  21  22  .  33   322  71   441  55    533   B1    553   77    933
            31     51   421  332  522  442   722   444   733   D1    B22
                   321       422  531  721   731   552   751   B21   B31
                             521       4321  4322  4332  931   4433  4443
                                             5321  4431  4432  5441  5442
                                                   5322  5332  6332  5532
                                                   5421  5422  7322  6621
                                                   6321  6322  7421  7332
                                                         7321        7422
                                                                     7521
                                                                     8421
                                                                     9321
                                                                     54321

Crossrefs

For no choices we have A370320, complement A239312.
The version for prime factors (not all divisors) is A370594, ranks A370647.
For multiple choices we have A370803, ranks A370811.
These partitions have ranks A370810.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
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.
A370592 counts partitions with choosable prime factors, ranks A368100.
A370593 counts partitions without choosable prime factors, ranks A355529.
A370804 counts non-condensed partitions with no ones, complement A370805.
A370814 counts factorizations with choosable divisors, complement A370813.
Cf. A355535, A355739, A355740, A367867, A367904, A368110, A370583, A370584, A370806, A370808.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]==1&]],{n,0,30}]

Extensions

More terms from Jinyuan Wang, Feb 14 2025

A371165 Positive integers with as many divisors (A000005) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

3, 5, 11, 17, 26, 31, 35, 38, 39, 41, 49, 57, 58, 59, 65, 67, 69, 77, 83, 86, 87, 94, 109, 119, 127, 129, 133, 146, 148, 157, 158, 179, 191, 202, 206, 211, 217, 235, 237, 241, 244, 253, 274, 277, 278, 283, 284, 287, 291, 298, 303, 319, 326, 331, 333, 334, 353
Offset: 1

Views

Author

Gus Wiseman, Mar 14 2024

Keywords

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.

Examples

			The terms together with their prime indices begin:
     3: {2}        67: {19}        158: {1,22}
     5: {3}        69: {2,9}       179: {41}
    11: {5}        77: {4,5}       191: {43}
    17: {7}        83: {23}        202: {1,26}
    26: {1,6}      86: {1,14}      206: {1,27}
    31: {11}       87: {2,10}      211: {47}
    35: {3,4}      94: {1,15}      217: {4,11}
    38: {1,8}     109: {29}        235: {3,15}
    39: {2,6}     119: {4,7}       237: {2,22}
    41: {13}      127: {31}        241: {53}
    49: {4,4}     129: {2,14}      244: {1,1,18}
    57: {2,8}     133: {4,8}       253: {5,9}
    58: {1,10}    146: {1,21}      274: {1,33}
    59: {17}      148: {1,1,12}    277: {59}
    65: {3,6}     157: {37}        278: {1,34}

Crossrefs

For prime factors instead of divisors on both sides we get A319899.
For prime factors on LHS we get A370802, for distinct prime factors A371177.
The RHS is A370820, for prime factors instead of divisors A303975.
For (greater than) instead of (equal) we get A371166.
For (less than) instead of (equal) we get A371167.
Partitions of this type are counted by A371172.
Other inequalities: A370348 (A371171), A371168 (A371173), 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.
A370814 counts divisor-choosable factorizations, complement A370813.
Cf. A000792, A003963, A355529, A355739, A355741, A368100, A370808, A371127.

Programs

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

Formula

A000005(a(n)) = A370820(a(n)).

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

Views

Author

Gus Wiseman, Mar 16 2024

Keywords

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.

Examples

			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}

Crossrefs

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.

Programs

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

Formula

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

A371173 Number of 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, 6, 7, 11, 11, 17, 20, 26, 34, 44, 56, 67, 84, 102, 131, 156, 195, 232, 283, 346, 411, 506, 598, 721, 855, 1025, 1204, 1448, 1689, 2018, 2363, 2796, 3265, 3840, 4489, 5242, 6104, 7106, 8280, 9595, 11143, 12862, 14926, 17197, 19862, 22841
Offset: 0

Views

Author

Gus Wiseman, Mar 16 2024

Keywords

Comments

The Heinz numbers of these partitions are given by A371168.

Examples

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

Crossrefs

The RHS is represented by A370820.
The version for equality is A371130 (ranks A370802), strict A371128.
For submultisets instead of parts on the LHS we get ranks A371166.
These partitions are ranked by A371168.
The opposite version is A371171, ranks A370348.
A000005 counts divisors.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
Cf. A003963, A319055, A355739, A370803, A370808, A370809, A370813, A370814.

Programs

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

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

Views

Author

Gus Wiseman, Jul 20 2022

Keywords

Examples

			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}

Crossrefs

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.

Programs

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

Formula

a(n) = A355733(A070826(n)).
a(p) = 2*a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022

Extensions

a(15)-a(21) from Michael S. Branicky, Aug 03 2022
a(22)-a(23) from Michael S. Branicky, Aug 08 2022

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

Views

Author

Gus Wiseman, Mar 05 2024

Keywords

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.

Examples

			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}

Crossrefs

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.

Programs

  • Mathematica
    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&]

A356065 Squarefree numbers whose prime indices are all prime-powers.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 21, 23, 31, 33, 35, 41, 51, 53, 55, 57, 59, 67, 69, 77, 83, 85, 93, 95, 97, 103, 105, 109, 115, 119, 123, 127, 131, 133, 155, 157, 159, 161, 165, 177, 179, 187, 191, 201, 205, 209, 211, 217, 227, 231, 241, 249, 253, 255, 265, 277
Offset: 1

Views

Author

Gus Wiseman, Jul 25 2022

Keywords

Examples

			105 has prime indices {2,3,4}, all three of which are prime-powers, so 105 is in the sequence.

Crossrefs

The multiplicative version (factorizations) is A050361, non-strict A000688.
Heinz numbers of the partitions counted by A054685, with 1's A106244, non-strict A023894, non-strict with 1's A023893.
Counting twice-partitions of this type gives A279786, non-strict A279784.
Counting twice-factorizations gives A295935, non-strict A296131.
These are the odd products of distinct elements of A302493.
Allowing prime index 1 gives A302496, non-strict A302492.
The case of primes (instead of prime-powers) is A302590, non-strict A076610.
These are the squarefree positions of 1's in A355741.
This is the squarefree case of A355743, complement A356066.
A001222 counts prime-power divisors.
A005117 lists the squarefree numbers.
A034699 gives maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A001970, A055887, A063834, A302601, A355731, A355744, A356064.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SquareFreeQ[#]&&And@@PrimePowerQ/@primeMS[#]&]

Formula

Intersection of A005117 and A355743.

A371172 Number of integer partitions of n with as many submultisets as distinct divisors of parts.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 3, 2, 3, 1, 4, 2, 1, 2, 3, 4, 2, 4, 1, 5, 2, 7, 5, 9, 4, 9, 15, 18, 16, 24, 13, 17, 23, 23, 22, 34, 17, 30, 31, 36, 29, 43, 21, 30, 35, 44, 28, 47, 19, 44
Offset: 0

Views

Author

Gus Wiseman, Mar 16 2024

Keywords

Comments

The Heinz numbers of these partitions are given by A371165.

Examples

			The partition (8,6,6) has 6 submultisets {(8,6,6),(8,6),(6,6),(8),(6),()} and 6 distinct divisors of parts {1,2,3,4,6,8}, so is counted under a(20).
The a(17) = 2 through a(24) = 9 partitions:
  (17)    (9,9)     (19)  (11,9)    (14,7)  (13,9)    (23)       (21,3)
  (13,4)  (15,3)          (15,5)    (17,4)  (21,1)    (19,4)     (22,2)
          (6,6,6)         (8,6,6)           (8,8,6)   (22,1)     (8,8,8)
          (12,3,3)        (12,4,4)          (10,6,6)  (15,4,4)   (10,8,6)
                          (18,1,1)          (16,3,3)  (12,10,1)  (12,6,6)
                                            (18,2,2)             (12,7,5)
                                            (20,1,1)             (18,3,3)
                                                                 (20,2,2)
                                                                 (12,10,2)

Crossrefs

The RHS is represented by A370820.
Counting parts on the LHS gives A371130 (ranks A370802), strict A371128.
These partitions are ranked by A371165.
A000005 counts divisors.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
Cf. A003963, A319055, A355739, A370803, A370808, A370813, A370814, A371166.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Length[Divisors[Times@@Prime/@#]] == Length[Union@@Divisors/@#]&]],{n,0,30}]

A355746 Number of different multisets that can be obtained by choosing a prime index (or a prime factor) of each integer from 2 to n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 6, 6, 12, 20, 20, 20, 26, 26, 36, 58, 116, 116, 140, 140, 280, 280, 384, 384, 536, 536, 536, 844, 1688, 2380, 2716, 2716, 5432, 8484, 10152, 10152, 13308, 13308, 18064, 21616, 43232, 43232, 47648, 47648, 54656, 84480, 114304, 114304
Offset: 1

Views

Author

Gus Wiseman, Jul 20 2022

Keywords

Examples

			The a(n) multisets for n = 2, 6, 10, 12:
  {1}  {1,1,1,2,3}  {1,1,1,1,1,2,2,3,4}  {1,1,1,1,1,1,2,2,3,4,5}
       {1,1,2,2,3}  {1,1,1,1,2,2,2,3,4}  {1,1,1,1,1,2,2,2,3,4,5}
                    {1,1,1,1,2,2,3,3,4}  {1,1,1,1,1,2,2,3,3,4,5}
                    {1,1,1,2,2,2,3,3,4}  {1,1,1,1,2,2,2,2,3,4,5}
                                         {1,1,1,1,2,2,2,3,3,4,5}
                                         {1,1,1,2,2,2,2,3,3,4,5}

Links

Crossrefs

The sum of the same integers is A000096.
The product of the same integers is A000142, Heinz number A070826.
The integers themselves are the rows of A131818 (shifted).
Counting sequences instead of multisets: A355537, with multiplicity A327486.
Using prime indices instead of 2..n gives A355744, for sequences A355741.
The version for divisors instead of prime factors is A355747.
A000040 lists the prime numbers.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A002110, A076610, A355538, A355731, A355733, A355740, A355742, A355745.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Tuples[primeMS/@Range[2,n]]]],{n,15}]
  • Python
    from sympy import factorint
from itertools import count, islice
def agen():
    s = {(1,)}
    for n in count(2):
        yield len(s)
        s = set(tuple(sorted(t+(d,))) for t in s for d in factorint(n))
print(list(islice(agen(), 53))) # Michael S. Branicky, Aug 03 2022

Formula

a(n) = A355744(A070826(n)).
a(p) = a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022

Extensions

a(28) and beyond from Michael S. Branicky, Aug 03 2022

A371127 Powers of 2 times powers > 1 of a prime-indexed prime number.

Original entry on oeis.org

3, 5, 6, 9, 10, 11, 12, 17, 18, 20, 22, 24, 25, 27, 31, 34, 36, 40, 41, 44, 48, 50, 54, 59, 62, 67, 68, 72, 80, 81, 82, 83, 88, 96, 100, 108, 109, 118, 121, 124, 125, 127, 134, 136, 144, 157, 160, 162, 164, 166, 176, 179, 191, 192, 200, 211, 216, 218, 236, 241
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2024

Keywords

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.

Examples

			The terms together with their prime indices begin:
      3: {2}
      5: {3}
      6: {1,2}
      9: {2,2}
     10: {1,3}
     11: {5}
     12: {1,1,2}
     17: {7}
     18: {1,2,2}
     20: {1,1,3}
     22: {1,5}
     24: {1,1,1,2}
     25: {3,3}
     27: {2,2,2}
     31: {11}
     34: {1,7}
     36: {1,1,2,2}

Crossrefs

Subset of A302540.
Subset of A336101 = powers of 2 times powers of primes.
Positions of 2's in A370820.
Counting prime factors instead of divisors gives A371287.
A000005 counts divisors.
A000961 lists powers of primes, A302596 of prime index.
A001221 counts distinct prime factors.
A003963 gives product of prime indices.
A027746 lists prime factors, indices A112798, length A001222.
A076610 lists products of primes of prime index.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000079, A005179, A007416, A303975, A319899, A355739, A370348, A370802, A371165-A371170, A371178.

Programs

  • Mathematica
    Select[Range[100],Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]==2&]
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