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.

Showing 1-8 of 8 results.

A383015 Numbers whose prime indices have more than one permutation with all equal run-sums.

Original entry on oeis.org

12, 40, 63, 112, 144, 325, 351, 352, 675, 832, 931, 1008, 1539, 1600, 1728, 2176, 2875, 3509, 3969, 4864, 6253, 7047, 7056, 8775, 9072, 11776, 12427, 12544, 12691, 16128, 19133, 20736, 20800, 22464, 23125, 26973, 29403, 29696, 32269, 43200, 49392, 57967, 59711
Offset: 1

Views

Author

Gus Wiseman, Apr 14 2025

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, sum A056239.
All terms appear to have even sum of prime indices.

Examples

			The terms together with their prime indices begin:
     12: {1,1,2}
     40: {1,1,1,3}
     63: {2,2,4}
    112: {1,1,1,1,4}
    144: {1,1,1,1,2,2}
    325: {3,3,6}
    351: {2,2,2,6}
    352: {1,1,1,1,1,5}
    675: {2,2,2,3,3}
    832: {1,1,1,1,1,1,6}
    931: {4,4,8}
   1008: {1,1,1,1,2,2,4}
   1539: {2,2,2,2,8}
   1600: {1,1,1,1,1,1,3,3}
   1728: {1,1,1,1,1,1,2,2,2}

Crossrefs

Compositions of this type are counted by A353851, ranked by A353848.
Positions of terms > 1 in A382877, zeros A383100 (complement A383014).
For run-lengths instead of sums we have A383089, counted by A383090.
The complement for run-lengths instead of sums is A383091, counted by A383092
Partitions of this type are counted by A383097.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A304442 counts compositions with equal run-sums, complement A382076.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A353837 counts partitions with distinct run-sums, ranks A353838.
A353847 gives composition run-sum transformation, for partitions A353832.
A353932 lists run-sums of standard compositions.
Cf. A000720, A000961, A001221, A001222, A329738, A353833, A354584, A381636, A381871, A382857, A382876, A382879.

Programs

  • Mathematica
    Select[Range[100],Length[Select[Permutations[PrimePi/@Join@@ConstantArray@@@FactorInteger[#]],SameQ@@Total/@Split[#]&]]>1&]

A383099 Numbers whose prime indices have exactly one permutation with all equal run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 48, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193
Offset: 1

Views

Author

Gus Wiseman, Apr 20 2025

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, sum A056239.

Examples

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   37: {12}
   41: {13}

Crossrefs

For distinct instead of equal run-sums we have A000961, counted by A000005.
These are the positions of 1 in A382877.
For more than one choice we have A383015.
Partitions of this type are counted by A383095.
For no choices we have A383100, counted by A383096.
For at least one choice we have A383110, counted by A383098, see A383013.
For run-lengths instead of sums we have A383112 = positions of 1 in A382857.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A353851 counts compositions with equal run-sums, ranks A353848.
Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
Cf. A000720, A001221, A001222, A351294, A353832, A353838, A353932, A354584, A382876, A383091.

Programs

  • Mathematica
    Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]==1&]

Formula

The complement is A383015 \/ A383100, for run-lengths A382879 \/ A383089.

A383089 Numbers whose prime indices have more than one permutation with all equal run-lengths.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140
Offset: 1

Views

Author

Gus Wiseman, Apr 18 2025

Keywords

Comments

First differs from A362606 (complement A359178 with 1) in having 180 and lacking 240.
First differs from A130092 (complement A130091) in having 360 and lacking 240.
First differs from A351295 (complement A351294) in having 216 and lacking 240.
Includes all squarefree numbers A005117 except the primes A000040.
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, sum A056239.

Examples

			The prime indices of 36 are {1,1,2,2}, and we have 4 permutations each having all equal run-lengths: (1,1,2,2), (1,2,1,2), (2,2,1,1), (2,1,2,1), so 36 is in the sequence.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}
   46: {1,9}
   51: {2,7}
   55: {3,5}
   57: {2,8}
   58: {1,10}
   60: {1,1,2,3}

Crossrefs

Positions of terms > 1 in A382857 (distinct A382771), zeros A382879, ones A383112.
For run-sums instead of lengths we have A383015, counted by A383097.
Partitions of this type are counted by A383090.
The complement is A383091, counted by A383092, just zero A382915, just one A383094.
For distinct instead of equal run-sums we have A383113.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A047966 counts partitions with equal run-lengths, compositions A329738.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct run-lengths, ranks A130091.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A000961, A001221, A001222, A048767, A353744, A353833, A381541, A381871, A382877, A383014, A383100.

Programs

  • Mathematica
    Select[Range[100],Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Length/@Split[#]&]]>1&]

Formula

The complement is A383091 = A382879 \/ A383112, counted by A382915 + A383094.

A383094 Number of integer partitions of n having exactly one permutation with all equal run-lengths.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 5, 6, 9, 7, 11, 10, 13, 12, 17, 14, 21, 16, 21, 18, 27, 22, 29, 22, 34, 25, 35, 28, 41, 28, 43, 30, 48, 38, 47, 38, 55, 36, 53, 46, 64, 40, 67, 42, 69, 54, 65, 46, 84, 51, 75, 62, 83, 52, 86, 62, 94, 70, 83, 58, 111, 60, 89, 80, 106, 74, 115, 66, 111
Offset: 0

Views

Author

Gus Wiseman, Apr 20 2025

Keywords

Examples

			The partition (222211) has exactly one permutation with all equal run-lengths: (221122), so is counted under a(10).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (11111)  (411)     (511)      (422)
                                     (111111)  (22111)    (611)
                                               (1111111)  (2222)
                                                          (22211)
                                                          (221111)
                                                          (11111111)

Crossrefs

The complement is ranked by A382879 \/ A383089.
For no choices we have A382915, ranks A382879.
For at least one choice we have A383013, for run-sums A383098, ranks A383110.
For more than one choice we have A383090, ranks A383089.
For at most one choice we have A383092, ranks A383091.
For run-sums instead of lengths we have A383095, ranks A383099.
Partitions of this type are ranked by A383112 = positions of 1 in A382857.
Cf. A047966, A072774, A098859, A100471, A100881, A100882, A100883, A304442.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A047993, A362608, A381871, A382076, A382771, A383096, A383097, A383111.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Length/@Split[#]&]]==1&]],{n,0,20}]

Extensions

More terms from Bert Dobbelaere, Apr 26 2025

A383090 Number of integer partitions of n having more than one permutation with all equal run-lengths.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 20, 28, 43, 55, 77, 107, 141, 183, 244, 312, 411, 521, 664, 837, 1069, 1328, 1667, 2069, 2578, 3166, 3929, 4791, 5895, 7168, 8749, 10594, 12883, 15500, 18741, 22493, 27069, 32334, 38760, 46133, 55065, 65367, 77686, 91905, 108927, 128431, 151674
Offset: 0

Views

Author

Gus Wiseman, Apr 19 2025

Keywords

Examples

			The partition (3322221) has 3 permutations with all equal run-lengths: (2323212), (2321232), (2123232), so is counted under a(15).
The partition (3322111111) has 2 permutations with all equal run-lengths: (1133112211), (1122113311), so is counted under a(16).
The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)
              (41)  (51)    (52)    (62)     (63)
                    (321)   (61)    (71)     (72)
                    (2211)  (421)   (431)    (81)
                            (3211)  (521)    (432)
                                    (3221)   (531)
                                    (3311)   (621)
                                    (4211)   (3321)
                                    (32111)  (4221)
                                             (4311)
                                             (5211)
                                             (32211)
                                             (42111)
                                             (222111)

Crossrefs

For no choices we have A382915, ranks A382879.
For at least one choice we have A383013, for run-sums A383098, ranks A383110.
Partitions of this type are ranked by A383089 = positions of terms > 1 in A382857.
The complement is A383091, counted by A383092.
For a unique choice we have A383094, ranks A383112.
The complement for run-sums is A383095 + A383096, ranks A383099 \/ A383100.
For run-sums we have A383097, ranked by A383015 = positions of terms > 1 in A382877.
For distinct instead of equal run-lengths we have A383111, ranks A383113.
Cf. A047966, A072774, A098859, A304442, A100471, A100881, A100882, A100883.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A047993, A237984, A329739, A381541, A381871, A382076, A382771.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Length/@Split[#]&]]>1&]],{n,0,15}]

Formula

The complement is counted by A383094 + A382915, ranks A383112 \/ A382879.

Extensions

More terms from Bert Dobbelaere, Apr 26 2025

A383112 Numbers whose multiset of prime indices has exactly one permutation with all equal run-lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 81, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 113, 116, 117, 121, 124, 125, 127
Offset: 1

Views

Author

Gus Wiseman, Apr 18 2025

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, sum A056239.
Includes all prime powers A000961.
Are there any terms x such that A001221(x) > 2?

Examples

			The prime indices of 144 are {1,1,1,1,2,2}, of which the only permutation with all equal run-lengths is (1,1,2,2,1,1), so 144 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  28: {1,1,4}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}

Crossrefs

These are the positions of 1 in A382857, distinct A382771.
The complement is A382879 \/ A383089, counted by A382915 + A383090.
For at most one permutation we have A383091, counted by A383092.
Partitions of this type are counted by A383094.
For run-sums instead of lengths we have A383099, counted by A383095.
A047966 counts partitions with equal run-lengths, ranks A072774.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct run-lengths, ranks A130091.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
Cf. A000961, A001221, A001222, A048767, A351294, A351295, A353833, A381434, A381540, A382877, A383100.

Programs

  • Mathematica
    Select[Range[100], Length[Select[Permutations[Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Length/@Split[#]&]]==1&]

A383092 Number of integer partitions of n having at most one permutation with all equal run-lengths.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 13, 16, 22, 28, 34, 46, 58, 69, 90, 114, 141, 178, 216, 271, 338, 418, 506, 630, 769, 941, 1140, 1399, 1675, 2051, 2454, 2975, 3561, 4289, 5094, 6137, 7274, 8692, 10269, 12249, 14414, 17128, 20110, 23767, 27872, 32849, 38346, 45094, 52552, 61533
Offset: 0

Views

Author

Gus Wiseman, Apr 19 2025

Keywords

Examples

			The partition (222211) has 1 permutation with all equal run-lengths: (221122), so is counted under a(10).
The partition (33211111) has no permutation with all equal run-lengths, so is counted under a(13).
The a(1) = 1 through a(7) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (221)    (33)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (511)
                            (11111)  (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

Crossrefs

For no choices we have A382915, ranks A382879.
For at least one choice we have A383013, for run-sums A383098, ranks A383110.
The complement is A383090, ranks A383089.
Partitions of this type are ranked by A383091 = positions of terms <= 1 in A382857.
For a unique choice we have A383094, ranks A383112.
For run-sums instead of lengths we have A383095 + A383096, ranks A383099 \/ A383100.
The complement for run-sums is A383097, ranks A383015, positions of terms > 1 in A382877.
Cf. A047966, A072774, A098859, A100471, A100881, A100882, A100883, A304442.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A006171, A047993, A362608, A381871, A382076, A382771, A383111.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Length/@Split[#]&]]<=1&]],{n,0,15}]

Formula

a(n) = A382915(n) + A383094(n).

Extensions

More terms from Bert Dobbelaere, Apr 26 2025

A383113 Numbers whose prime indices have more than one permutation with all distinct run-lengths.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 216, 224, 232, 236, 242
Offset: 1

Views

Author

Gus Wiseman, Apr 20 2025

Keywords

Comments

First differs from A177425, A182854, A367589 in having 216.
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 360 are {1,1,1,2,2,3}, with six permutations with all distinct run-lengths:
  (1,1,1,2,2,3)
  (1,1,1,3,2,2)
  (2,2,1,1,1,3)
  (2,2,3,1,1,1)
  (3,1,1,1,2,2)
  (3,2,2,1,1,1)
so 360 is in the sequence.
The terms together with their prime indices begin:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  28: {1,1,4}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
  75: {2,3,3}
  76: {1,1,8}
  80: {1,1,1,1,3}

Crossrefs

For exactly one permutation we have A000961, counted by A000005.
For no choices we have A351293, counted by A351295, conjugate A381433, equal A382879.
For at least one choice we have A351294, conjugate A381432, counted by A239455.
These are positions of terms > 1 in A382771, firsts A382772, equal A382878.
For equal run-lengths we have A383089, positions of terms > 1 in A382857.
Partitions of this type are counted by A383111.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct run-lengths (ordered A242882), ranks A130091.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
Cf. A000720, A001221, A001222, A047966, A048767, A351013, A351202, A381435, A382876, A383090, A383091, A383092, A383112.

Programs

  • Mathematica
    Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], UnsameQ@@Length/@Split[#]&]]>1&]

Formula

The complement is A000961 \/ A351293, counted by A000005 + A351295.
Showing 1-8 of 8 results.

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