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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A370348 Numbers k such that there are fewer divisors of prime indices of k than there are prime indices of k.

Original entry on oeis.org

4, 8, 12, 16, 18, 20, 24, 27, 32, 36, 40, 44, 48, 50, 54, 56, 60, 64, 68, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 124, 125, 126, 128, 132, 135, 136, 144, 150, 160, 162, 164, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 216, 220, 224, 225, 236, 240, 242, 243, 248, 250, 252, 256
Offset: 1

Views

Author

Robert Israel, Feb 15 2024

Keywords

Comments

No multiple of a term is a term of A368110.

Examples

			a(5) = 18 is a term because the prime indices of 18 = 2 * 3^2 are 1,2,2, and there are 3 of these but only 2 divisors of prime indices, namely 1 and 2.

Links

Crossrefs

The LHS is A370820, firsts A371131.
The version for equality is A370802, counted by A371130, strict A371128.
For submultisets instead of parts on the RHS we get A371167.
The opposite version is A371168, counted by A371173.
The weak version is A371169.
The complement is A371170.
Partitions of this type are counted by A371171.
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, A239312, A303975, A319899, A355529, A355739, A370808, A371127.

Programs

  • Maple
    filter:= proc(n) uses numtheory; local F,D,t;
   F:= map(t -> [pi(t[1]),t[2]], ifactors(n)[2]);
   D:= `union`(seq(divisors(t[1]), t = F));
   nops(D) < add(t[2], t = F)
end proc:
select(filter, [$1..300]);
  • Mathematica
    filter[n_] := Module[{F, d},
    F = {PrimePi[#[[1]]], #[[2]]}& /@ FactorInteger[n];
    d = Union[Flatten[Divisors /@ F[[All, 1]]]];
    Length[d] < Total[F[[All, 2]]]];
Select[Range[300], filter] (* Jean-François Alcover, Mar 08 2024, after Maple code *)

A371130 Number of integer partitions of n such that the number of parts is equal to the number of distinct divisors of parts.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 4, 2, 4, 5, 5, 11, 10, 16, 17, 21, 26, 32, 44, 53, 69, 71, 101, 110, 148, 168, 205, 249, 289, 356, 418, 502, 589, 716, 812, 999, 1137, 1365, 1566, 1873, 2158, 2537, 2942, 3449, 4001, 4613, 5380, 6193, 7220, 8224, 9575, 10926, 12683, 14430
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2024

Keywords

Comments

The Heinz numbers of these partitions are given by A370802.

Examples

			The partition (6,2,2,1) has 4 parts and 4 distinct divisors of parts {1,2,3,6} so is counted under a(11).
The a(1) = 1 through a(11) = 11 partitions:
  (1)  .  (21)  (22)  .  (33)   (322)  (71)   (441)   (55)    (533)
                (31)     (51)   (421)  (332)  (522)   (442)   (722)
                         (321)         (422)  (531)   (721)   (731)
                         (411)         (521)  (4311)  (4321)  (911)
                                              (6111)  (6211)  (4322)
                                                              (4331)
                                                              (5321)
                                                              (5411)
                                                              (6221)
                                                              (6311)
                                                              (8111)

Crossrefs

The LHS is represented by A001222, distinct A000021.
These partitions are ranked by A370802.
The RHS is represented by A370820, for prime factors A303975.
The strict case is A371128.
For (greater than) instead of (equal to) we have A371171, ranks A370348.
For submultisets instead of parts on the LHS we have A371172.
For (less than) instead of (equal to) we have A371173, ranked by A371168.
Counting only distinct parts on the LHS gives A371178, ranks A371177.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
Cf. A003963, A319055, A355731, A370803, A370808, A370809, A370813, A370814.

Programs

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

A371128 Number of strict integer partitions of n containing all distinct divisors of all parts.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 3, 5, 3, 5, 6, 7, 7, 8, 8, 9, 12, 13, 13, 14, 15, 16, 19, 23, 25, 26, 26, 27, 36, 37, 40, 42, 46, 50, 55, 66, 65, 71, 71, 82, 90, 102, 103, 114, 117, 130, 147, 154, 166, 176, 182, 194, 228, 239, 259, 267, 287, 307, 336
Offset: 0

Views

Author

Gus Wiseman, Mar 18 2024

Keywords

Comments

Also strict integer partitions such that the number of parts is equal to the number of distinct divisors of all parts.

Examples

			The a(9) = 1 through a(19) = 7 partitions (A..H = 10..17):
  531  721   731   B1    751   D1    B31    D21    B51    H1     B71
       4321  5321  5421  931   B21   7521   7531   D31    9531   D51
                   6321  7321  7421  8421   64321  B321   A521   B521
                                     9321          65321  B421   D321
                                     54321         74321  75321  75421
                                                          84321  76321
                                                                 94321

Crossrefs

The LHS is represented by A001221, distinct case of A001222.
The RHS is represented by A370820, for prime factors A303975.
Strict case of A371130 (ranks A370802) and A371178 (ranks A371177).
The complement is counted by A371180, non-strict A371132.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts partitions without divisors, strict A303362, ranks A316476.
Cf. A000837, A003963, A239312, A285573, A319055, A370803, A370808, A371171, A371172, A371173.

Programs

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

A371171 Number of integer partitions of n with more parts than distinct divisors of parts.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 5, 9, 12, 18, 26, 34, 50, 65, 92, 121, 161, 209, 274, 353, 456, 590, 745, 950, 1195, 1507, 1885, 2350, 2923, 3611, 4465, 5485, 6735, 8223, 10050, 12195, 14822, 17909, 21653, 26047, 31340, 37557, 44990, 53708, 64068, 76241, 90583, 107418
Offset: 1

Views

Author

Gus Wiseman, Mar 16 2024

Keywords

Comments

The Heinz numbers of these partitions are given by A370348.

Examples

			The partition (3,2,1,1) has 4 parts {1,2,3,4} and 3 distinct divisors of parts {1,2,3}, so is counted under a(7).
The a(0) = 0 through a(8) = 12 partitions:
  .  .  (11)  (111)  (211)   (221)    (222)     (331)      (2222)
                     (1111)  (311)    (2211)    (511)      (3221)
                             (2111)   (3111)    (2221)     (3311)
                             (11111)  (21111)   (3211)     (4211)
                                      (111111)  (4111)     (5111)
                                                (22111)    (22211)
                                                (31111)    (32111)
                                                (211111)   (41111)
                                                (1111111)  (221111)
                                                           (311111)
                                                           (2111111)
                                                           (11111111)

Crossrefs

The partitions are ranked by A370348.
The opposite version is A371173, ranked by A371168.
The RHS is represented by A370820, positions of twos A371127.
The version for equality is A371130 (ranks A370802), strict A371128.
For submultisets instead of parts on the LHS we get ranks A371167.
A000005 counts divisors.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
Cf. A003963, A319055, A355731, A370803, A370808, A370809, A370813, A370814.

Programs

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

A371177 Positive integers whose prime indices include all distinct divisors of all prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 30, 32, 34, 36, 40, 42, 44, 48, 50, 54, 60, 62, 64, 66, 68, 72, 80, 82, 84, 88, 90, 96, 100, 102, 108, 110, 118, 120, 124, 126, 128, 132, 134, 136, 144, 150, 160, 162, 164, 166, 168, 170, 176, 180, 186, 192, 198, 200
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.
Also positive integers with as many distinct prime factors (A001221) as distinct divisors of prime indices (A370820).

Examples

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}

Crossrefs

The LHS is A001221, distinct case of A001222.
The RHS is A370820, for prime factors A303975.
For bigomega on the LHS we have A370802, counted by A371130.
For divisors on the LHS we have A371165, counted by A371172.
Partitions of this type are counted by A371178, strict A371128.
The complement is A371179, counted by A371132.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts partitions without divisors, strict A303362, ranks A316476.
Cf. A000837, A003963, A239312, A285573, A355529, A370813, A371168.

Programs

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

Formula

A001221(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}]

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}]

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&]

A371178 Number of integer partitions of n containing all divisors of all parts.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 37, 48, 62, 80, 101, 127, 162, 202, 252, 312, 386, 475, 585, 713, 869, 1056, 1278, 1541, 1859, 2232, 2675, 3196, 3811, 4534, 5386, 6379, 7547, 8908, 10497, 12345, 14501, 16999, 19897, 23253, 27135, 31618, 36796, 42756
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2024

Keywords

Comments

The Heinz numbers of these partitions are given by A371177.
Also partitions such that the number of distinct parts is equal to the number of distinct divisors of parts.

Examples

			The partition (4,2,1,1) contains all distinct divisors {1,2,4}, so is counted under a(8).
The partition (4,4,3,2,2,2,1) contains all distinct divisors {1,2,3,4} so is counted under 4 + 4 + 3 + 2 + 2 + 2 + 1 = 18. - _David A. Corneth_, Mar 18 2024
The a(0) = 1 through a(8) = 12 partitions:
  ()  (1)  (11)  (21)   (31)    (221)    (51)      (331)      (71)
                 (111)  (211)   (311)    (321)     (421)      (521)
                        (1111)  (2111)   (2211)    (511)      (3221)
                                (11111)  (3111)    (2221)     (3311)
                                         (21111)   (3211)     (4211)
                                         (111111)  (22111)    (5111)
                                                   (31111)    (22211)
                                                   (211111)   (32111)
                                                   (1111111)  (221111)
                                                              (311111)
                                                              (2111111)
                                                              (11111111)

Crossrefs

The LHS is represented by A001221, distinct case of A001222.
For partitions with no divisors of parts we have A305148, ranks A316476.
The RHS is represented by A370820, for prime factors A303975.
The strict case is A371128.
Counting all parts on the LHS gives A371130, ranks A370802.
The complement is counted by A371132.
For submultisets instead of distinct parts we have A371172, ranks A371165.
These partitions have ranks A371177.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Cf. A000837, A003963, A239312, A285573, A305148, A319055, A355529, A370803, A370808, A370813, A371168, A371171, A371173.

Programs

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

A371288 Numbers whose distinct prime indices form the set of divisors of some positive integer.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 32, 34, 36, 40, 42, 44, 48, 50, 54, 62, 64, 68, 72, 80, 82, 84, 88, 96, 100, 108, 118, 124, 126, 128, 134, 136, 144, 160, 162, 164, 166, 168, 176, 192, 200, 216, 218, 230, 236, 242, 248, 250, 252, 254, 256, 268, 272, 288
Offset: 1

Views

Author

Gus Wiseman, Mar 22 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 694782 are {1,2,2,5,5,5,10} with distinct elements {1,2,5,10}, which form the set of divisors of 10, so 694782 is in the sequence.
The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}

Links

Crossrefs

The squarefree case is A371283, unsorted version A275700.
Partitions of this type are counted by A371284, strict A054973.
Products of squarefree terms are A371286, unsorted version A371285.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
Cf. A000720, A003963, A005179, A007416, A034729, A370820, A371131, A371177.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Union[prix[#]]==Divisors[Max@@prix[#]]&]
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