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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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[#]&]=={}&]

A383111 Number of integer partitions of n having more than one permutation with all distinct run-lengths.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 8, 9, 13, 17, 26, 27, 43, 51, 61, 78, 103, 115, 153, 174, 213, 255, 316, 354, 442, 508, 610, 701, 848, 950, 1153, 1303, 1539, 1750, 2075, 2318, 2738, 3081
Offset: 0

Views

Author

Gus Wiseman, Apr 20 2025

Keywords

Examples

			The partition (2,1,1) has two permutations with all distinct run-lengths: (1,1,2), (2,1,1), so it is counted under a(4).
The a(4) = 1 through a(9) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (5111)     (3222)
                          (4111)    (22211)    (6111)
                          (22111)   (41111)    (22221)
                          (31111)   (221111)   (33111)
                          (211111)  (311111)   (51111)
                                    (2111111)  (222111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

Crossrefs

For a unique choice we have A000005, ranks A000961.
For at least one choice we have A239455, ranks A351294, conjugate A381432.
For no choices we have A351293, ranks A351295, conjugate A381433.
The complement is A351293 + A000005, ranks too dense.
For equal instead of distinct run-lengths we have A383090, ranks A383089.
These partitions are ranked by A383113 = positions of terms > 1 in A382771.
Cf. A047966, A098859, A130091, A353837, A100471, A100881, A100882, A100883
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A329738 counts compositions with equal run-lengths, ranks A353744.
Cf. A047993, A329739, A381541, A381636, A381717, A382857, A382915, A383013, A383092, A383094, A383097.

Programs

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

Extensions

a(21)-a(38) from Jakub Buczak, May 04 2025
Previous Showing 11-14 of 14 results.

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