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A383112 Numbers whose multiset of prime indices has exactly one permutation with all equal run-lengths.

%I A383112 #7 Apr 21 2025 13:25:01
%S A383112 1,2,3,4,5,7,8,9,11,12,13,16,17,18,19,20,23,25,27,28,29,31,32,37,41,
%T A383112 43,44,45,47,49,50,52,53,59,61,63,64,67,68,71,72,73,75,76,79,81,83,89,
%U A383112 92,97,98,99,101,103,107,108,109,113,116,117,121,124,125,127
%N A383112 Numbers whose multiset of prime indices has exactly one permutation with all equal run-lengths.
%C A383112 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.
%C A383112 Includes all prime powers A000961.
%C A383112 Are there any terms x such that A001221(x) > 2?
%e A383112 The prime indices of 144 are {1,1,1,1,2,2}, of which the only permutation with all equal run-lengths is (1,1,2,2,1,1), so 144 is in the sequence.
%e A383112 The terms together with their prime indices begin:
%e A383112    1: {}
%e A383112    2: {1}
%e A383112    3: {2}
%e A383112    4: {1,1}
%e A383112    5: {3}
%e A383112    7: {4}
%e A383112    8: {1,1,1}
%e A383112    9: {2,2}
%e A383112   11: {5}
%e A383112   12: {1,1,2}
%e A383112   13: {6}
%e A383112   16: {1,1,1,1}
%e A383112   17: {7}
%e A383112   18: {1,2,2}
%e A383112   19: {8}
%e A383112   20: {1,1,3}
%e A383112   23: {9}
%e A383112   25: {3,3}
%e A383112   27: {2,2,2}
%e A383112   28: {1,1,4}
%e A383112   29: {10}
%e A383112   31: {11}
%e A383112   32: {1,1,1,1,1}
%t A383112 Select[Range[100], Length[Select[Permutations[Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Length/@Split[#]&]]==1&]
%Y A383112 These are the positions of 1 in A382857, distinct A382771.
%Y A383112 The complement is A382879 \/ A383089, counted by A382915 + A383090.
%Y A383112 For at most one permutation we have A383091, counted by A383092.
%Y A383112 Partitions of this type are counted by A383094.
%Y A383112 For run-sums instead of lengths we have A383099, counted by A383095.
%Y A383112 A047966 counts partitions with equal run-lengths, ranks A072774.
%Y A383112 A056239 adds up prime indices, row sums of A112798.
%Y A383112 A098859 counts partitions with distinct run-lengths, ranks A130091.
%Y A383112 A329738 counts compositions with equal run-lengths, ranks A353744.
%Y A383112 A329739 counts compositions with distinct run-lengths, ranks A351596.
%Y A383112 Cf. A000961, A001221, A001222, A048767, A351294, A351295, A353833, A381434, A381540, A382877, A383100.
%K A383112 nonn
%O A383112 1,2
%A A383112 _Gus Wiseman_, Apr 18 2025