A383113 -id:A383113 - cp's OEIS frontend

A382771 Number of ways to permute the prime indices of n so that the run-lengths are all different.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 2, 1, 2, 0, 2, 1, 2, 0, 2, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 2, 1, 0, 2, 2, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Apr 07 2025

nonn

The first x with a(x) > 0 but A382857(x) > 1 is a(216) = 4, A382857(216) = 4.

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.

			The a(96) = 4 permutations are:
  (1,1,1,1,1,2)
  (1,1,1,2,1,1)
  (1,1,2,1,1,1)
  (2,1,1,1,1,1)
The a(216) = 4 permutations are:
  (1,1,2,2,2,1)
  (1,2,2,2,1,1)
  (2,1,1,1,2,2)
  (2,2,1,1,1,2)
The a(360) = 6 permutations are:
  (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)

Positions of 1 are A000961.
Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433, equal A382879.
Sorted positions of first appearances are A382772, equal A382878.
For prescribed signature we have A382773, equal A382858.
The restriction to factorials is A382774, equal A335407.
For equal instead of distinct run-lengths we have A382857.
For run-sums instead of run-lengths we have A382876, equal A382877.
Positions of terms > 1 are A383113.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, complement A351293.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
Cf. A000720, A001221, A001222, A003242, A048767, A051903, A051904, A130091, A238130, A351013, A351202.

Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],UnsameQ@@Length/@Split[#]&]],{n,30}]

a(A181821(n)) = a(A304660(n)) = A382773(n).

a(n!) = A382774(n).

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

Gus Wiseman, Apr 18 2025

nonn

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.

			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}

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.

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

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

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

Gus Wiseman, Apr 19 2025

nonn

			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)

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.

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

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

More terms from Bert Dobbelaere, Apr 26 2025

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

Gus Wiseman, Apr 18 2025

nonn

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.

			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}

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.

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

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

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

Gus Wiseman, Apr 20 2025

			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)

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.

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

a(21)-a(38) from Jakub Buczak, May 04 2025

cp's OEIS Frontend

A382771 Number of ways to permute the prime indices of n so that the run-lengths are all different.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A383091 Numbers whose prime indices have at most one permutation with all equal run-lengths.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions