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A383113 Numbers whose prime indices have more than one permutation with all distinct run-lengths.

%I A383113 #5 Apr 22 2025 09:09:57
%S A383113 12,18,20,24,28,40,44,45,48,50,52,54,56,63,68,72,75,76,80,88,92,96,98,
%T A383113 99,104,108,112,116,117,124,135,136,144,147,148,152,153,160,162,164,
%U A383113 171,172,175,176,184,188,189,192,200,207,208,212,216,224,232,236,242
%N A383113 Numbers whose prime indices have more than one permutation with all distinct run-lengths.
%C A383113 First differs from A177425, A182854, A367589 in having 216.
%C A383113 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.
%F A383113 The complement is A000961 \/ A351293, counted by A000005 + A351295.
%e A383113 The prime indices of 360 are {1,1,1,2,2,3}, with six permutations with all distinct run-lengths:
%e A383113   (1,1,1,2,2,3)
%e A383113   (1,1,1,3,2,2)
%e A383113   (2,2,1,1,1,3)
%e A383113   (2,2,3,1,1,1)
%e A383113   (3,1,1,1,2,2)
%e A383113   (3,2,2,1,1,1)
%e A383113 so 360 is in the sequence.
%e A383113 The terms together with their prime indices begin:
%e A383113   12: {1,1,2}
%e A383113   18: {1,2,2}
%e A383113   20: {1,1,3}
%e A383113   24: {1,1,1,2}
%e A383113   28: {1,1,4}
%e A383113   40: {1,1,1,3}
%e A383113   44: {1,1,5}
%e A383113   45: {2,2,3}
%e A383113   48: {1,1,1,1,2}
%e A383113   50: {1,3,3}
%e A383113   52: {1,1,6}
%e A383113   54: {1,2,2,2}
%e A383113   56: {1,1,1,4}
%e A383113   63: {2,2,4}
%e A383113   68: {1,1,7}
%e A383113   72: {1,1,1,2,2}
%e A383113   75: {2,3,3}
%e A383113   76: {1,1,8}
%e A383113   80: {1,1,1,1,3}
%t A383113 Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], UnsameQ@@Length/@Split[#]&]]>1&]
%Y A383113 For exactly one permutation we have A000961, counted by A000005.
%Y A383113 For no choices we have A351293, counted by A351295, conjugate A381433, equal A382879.
%Y A383113 For at least one choice we have A351294, conjugate A381432, counted by A239455.
%Y A383113 These are positions of terms > 1 in A382771, firsts A382772, equal A382878.
%Y A383113 For equal run-lengths we have A383089, positions of terms > 1 in A382857.
%Y A383113 Partitions of this type are counted by A383111.
%Y A383113 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A383113 A056239 adds up prime indices, row sums of A112798.
%Y A383113 A098859 counts partitions with distinct run-lengths (ordered A242882), ranks A130091.
%Y A383113 A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
%Y A383113 Cf. A000720, A001221, A001222, A047966, A048767, A351013, A351202, A381435, A382876, A383090, A383091, A383092, A383112.
%K A383113 nonn
%O A383113 1,1
%A A383113 _Gus Wiseman_, Apr 20 2025