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A383099 Numbers whose prime indices have exactly one permutation with all equal run-sums.

%I A383099 #7 Apr 22 2025 08:04:24
%S A383099 1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,36,37,41,43,47,48,
%T A383099 49,53,59,61,64,67,71,73,79,81,83,89,97,101,103,107,109,113,121,125,
%U A383099 127,128,131,137,139,149,151,157,163,167,169,173,179,181,191,193
%N A383099 Numbers whose prime indices have exactly one permutation with all equal run-sums.
%C A383099 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 A383099 The complement is A383015 \/ A383100, for run-lengths A382879 \/ A383089.
%e A383099 The terms together with their prime indices begin:
%e A383099     1: {}
%e A383099     2: {1}
%e A383099     3: {2}
%e A383099     4: {1,1}
%e A383099     5: {3}
%e A383099     7: {4}
%e A383099     8: {1,1,1}
%e A383099     9: {2,2}
%e A383099    11: {5}
%e A383099    13: {6}
%e A383099    16: {1,1,1,1}
%e A383099    17: {7}
%e A383099    19: {8}
%e A383099    23: {9}
%e A383099    25: {3,3}
%e A383099    27: {2,2,2}
%e A383099    29: {10}
%e A383099    31: {11}
%e A383099    32: {1,1,1,1,1}
%e A383099    36: {1,1,2,2}
%e A383099    37: {12}
%e A383099    41: {13}
%t A383099 Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]==1&]
%Y A383099 For distinct instead of equal run-sums we have A000961, counted by A000005.
%Y A383099 These are the positions of 1 in A382877.
%Y A383099 For more than one choice we have A383015.
%Y A383099 Partitions of this type are counted by A383095.
%Y A383099 For no choices we have A383100, counted by A383096.
%Y A383099 For at least one choice we have A383110, counted by A383098, see A383013.
%Y A383099 For run-lengths instead of sums we have A383112 = positions of 1 in A382857.
%Y A383099 A056239 adds up prime indices, row sums of A112798.
%Y A383099 A304442 counts partitions with equal run-sums, ranks A353833.
%Y A383099 A353851 counts compositions with equal run-sums, ranks A353848.
%Y A383099 Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
%Y A383099 Cf. A000720, A001221, A001222, A351294, A353832, A353838, A353932, A354584, A382876, A383091.
%K A383099 nonn
%O A383099 1,2
%A A383099 _Gus Wiseman_, Apr 20 2025