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.

Previous Showing 11-19 of 19 results.

A383110 Numbers whose prime indices have a permutation with all equal run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173
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 prime indices of 144 are {1,1,1,1,2,2}, with permutations with equal run sums (1,1,1,1,2,2), (1,1,2,1,1,2), (2,1,1,2,1,1), (2,2,1,1,1,1), 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}
  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}

Crossrefs

For distinct run-sums we appear to have complement of A381636 (counted by A381717).
These are the positions of positive terms in A382877.
For run-lengths instead of sums we have complement of A382879, counted by A383013.
For more than one choice we have A383015.
Partitions of this type are counted by A383098.
For a unique choice we have A383099, counted by A383095.
The complement is A383100, counted by A383096.
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. A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383097.

Programs

  • Mathematica
    Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]>0&]

Formula

Equals A383015 \/ A383099, counted by A353851 \/ A383095.

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.

A383096 Number of integer partitions of n having no permutation with all equal run-sums.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 4, 13, 15, 25, 35, 54, 58, 99, 128, 168, 217, 295, 358, 488, 603, 784, 995, 1253, 1517, 1953, 2429, 2997, 3688, 4563, 5532, 6840, 8311, 10135, 12303, 14875, 17842, 21635, 26008, 31177, 37247, 44581, 53062, 63259, 75130, 89096, 105551, 124752, 147015, 173520
Offset: 0

Views

Author

Gus Wiseman, Apr 17 2025

Keywords

Examples

			The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (22211)
                             (211111)  (32111)
                                       (311111)
                                       (2111111)

Crossrefs

For distinct instead of equal run-sums we appear to have A381717, q.v.
For run-lengths instead of sums we have A382915, ranks A382879, by signature A382914.
For more than one permutation we have A383097, ranks A383015.
The complement is counted by A383098, ranks A383110
These partitions are ranked by A383100, positions of 0 in A382877.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
A383095 counts partitions having a unique permutation with equal run-sums, ranks A383099.
Cf. A006171, A329738, A353832, A353839, A353932, A354584, A382076, A382857, A383094.

Programs

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

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&]

A383091 Numbers whose prime indices have at most 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, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109
Offset: 1

Views

Author

Gus Wiseman, Apr 18 2025

Keywords

Comments

First differs from A359178 (complement A362606) in having 1, 240 and lacking 180.
First differs from A130091 (complement A130092) in having 240 and lacking 360.
First differs from A351294 (complement A351295) in having 240 and lacking 216.
Includes all primes A000040 and prime powers A000961.
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 144 are {1,1,1,1,2,2}, with just one permutation with all equal run-lengths (1,1,2,2,1,1), so 144 is in the sequence.
The prime indices of 240 are {1,1,1,1,2,3}, which have no permutation with all equal run-lengths, so 240 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}
  24: {1,1,1,2}

Crossrefs

These are positions of zeros and ones in A382857, just zeros A382879, just ones A383112.
The complement for run-sums instead of lengths is A383015, counted by A383097.
The complement is A383089, counted by A383090.
Partitions of this type are counted by A383092, just zero A382915, just one A383094.
For run-sums instead of lengths we have A383099 \/ A383100, counted by A383095 + A383096.
A047966 counts partitions with equal run-lengths, compositions A329738.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct run-lengths, ranks A130091.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
Cf. A000720, A000961, A001221, A001222, A048767, A351294, A353744, A353833, A381435, A382771, A382877, A383113.

Programs

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

Formula

Equals A382879 \/ A383112, counted by A382915 + A383094.

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

A382914 Numbers k such that it is not possible to permute a multiset whose multiplicities are the prime indices of k so that the run-lengths are all equal.

Original entry on oeis.org

10, 14, 22, 26, 28, 33, 34, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 66, 68, 69, 74, 76, 78, 82, 85, 86, 87, 88, 92, 93, 94, 95, 102, 104, 106, 111, 114, 115, 116, 118, 119, 122, 123, 124, 129, 130, 134, 136, 138, 141, 142, 145, 146, 148, 152, 153, 155, 156
Offset: 1

Views

Author

Gus Wiseman, Apr 09 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:
  10: {1,3}
  14: {1,4}
  22: {1,5}
  26: {1,6}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
  38: {1,8}
  39: {2,6}
  44: {1,1,5}
  46: {1,9}
  51: {2,7}
  52: {1,1,6}
  55: {3,5}
  57: {2,8}
  58: {1,10}
  62: {1,11}
  66: {1,2,5}

Crossrefs

For anti-run permutations we have A335126, complement A335127.
Zeros of A382858, anti-run A335125.
For prime indices instead of signature we have A382879, counted by A382915.
For distinct run-lengths we have A382912 (zeros of A382773), complement A382913.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798.
A140690 lists numbers whose binary expansion has equal run-lengths, distinct A044813.
A304442 counts partitions with equal run-sums, ranks A353833.
A164707 lists numbers whose binary form has equal runs of ones, distinct A328592.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
Cf. A382857 (firsts A382878), A382771 (firsts A382772).
Cf. A000720, A000961, A001221, A001222, A048767, A181821, A305936, A351013, A382877.

Programs

  • Mathematica
    nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Select[Permutations[nrmptn[#]],SameQ@@Length/@Split[#]&]=={}&]

A383088 Numbers whose multiset of prime indices does not have all equal run-sums.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105
Offset: 1

Views

Author

Gus Wiseman, Apr 17 2025

Keywords

Comments

First differs from A381871 in having 36.
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}, with run-sums (2,4), so 36 is in the sequence, even though we have the multiset partition {{1,1},{2},{2}} with equal sums.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   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}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}

Crossrefs

For run-lengths instead of sums we have A059404, distinct A130092.
The complement is A353833, counted by A304442.
For distinct instead of equal run-sums we have A353839.
Partitions of this type are counted by A382076.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with a common run-sum, ranks A353848.
A353862 gives the greatest run-sum of prime indices, least A353931.
A382877 counts permutations of prime indices with equal run-sums, zeros A383100.
A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.
Cf. A000720, A006171, A300273, A353861, A353932, A354584, A383014, A383015, A383095, A383097, A383099.

Programs

  • Mathematica
    Select[Range[100], !SameQ@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&]
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