cp's OEIS Frontend

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

%I A383089 #9 Apr 19 2025 19:34:45
%S A383089 6,10,14,15,21,22,26,30,33,34,35,36,38,39,42,46,51,55,57,58,60,62,65,
%T A383089 66,69,70,74,77,78,82,84,85,86,87,90,91,93,94,95,100,102,105,106,110,
%U A383089 111,114,115,118,119,120,122,123,126,129,130,132,133,134,138,140
%N A383089 Numbers whose prime indices have more than one permutation with all equal run-lengths.
%C A383089 First differs from A362606 (complement A359178 with 1) in having 180 and lacking 240.
%C A383089 First differs from A130092 (complement A130091) in having 360 and lacking 240.
%C A383089 First differs from A351295 (complement A351294) in having 216 and lacking 240.
%C A383089 Includes all squarefree numbers A005117 except the primes A000040.
%C A383089 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.
%F A383089 The complement is A383091 = A382879 \/ A383112, counted by A382915 + A383094.
%e A383089 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.
%e A383089 The terms together with their prime indices begin:
%e A383089     6: {1,2}
%e A383089    10: {1,3}
%e A383089    14: {1,4}
%e A383089    15: {2,3}
%e A383089    21: {2,4}
%e A383089    22: {1,5}
%e A383089    26: {1,6}
%e A383089    30: {1,2,3}
%e A383089    33: {2,5}
%e A383089    34: {1,7}
%e A383089    35: {3,4}
%e A383089    36: {1,1,2,2}
%e A383089    38: {1,8}
%e A383089    39: {2,6}
%e A383089    42: {1,2,4}
%e A383089    46: {1,9}
%e A383089    51: {2,7}
%e A383089    55: {3,5}
%e A383089    57: {2,8}
%e A383089    58: {1,10}
%e A383089    60: {1,1,2,3}
%t A383089 Select[Range[100],Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Length/@Split[#]&]]>1&]
%Y A383089 Positions of terms > 1 in A382857 (distinct A382771), zeros A382879, ones A383112.
%Y A383089 For run-sums instead of lengths we have A383015, counted by A383097.
%Y A383089 Partitions of this type are counted by A383090.
%Y A383089 The complement is A383091, counted by A383092, just zero A382915, just one A383094.
%Y A383089 For distinct instead of equal run-sums we have A383113.
%Y A383089 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A383089 A047966 counts partitions with equal run-lengths, compositions A329738.
%Y A383089 A055396 gives least prime index, greatest A061395.
%Y A383089 A056239 adds up prime indices, row sums of A112798.
%Y A383089 A098859 counts partitions with distinct run-lengths, ranks A130091.
%Y A383089 A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
%Y A383089 A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
%Y A383089 A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
%Y A383089 Cf. A000720, A000961, A001221, A001222, A048767, A353744, A353833, A381541, A381871, A382877, A383014, A383100.
%K A383089 nonn
%O A383089 1,1
%A A383089 _Gus Wiseman_, Apr 18 2025