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

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

1,1

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.

Links

Table of n, a(n) for n=1..60.

Formula

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

Example

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}

Mathematica

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

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.

Sequence in context: A119899 A351295 A362606 * A130092 A289619 A329140

Adjacent sequences: A383086 A383087 A383088 * A383090 A383091 A383092

Keyword

nonn

Author

Gus Wiseman, Apr 18 2025

Status

approved