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A383100 Numbers whose prime indices have no permutation with all equal run-sums.

%I A383100 #6 Apr 22 2025 08:05:54
%S A383100 6,10,14,15,18,20,21,22,24,26,28,30,33,34,35,38,39,42,44,45,46,50,51,
%T A383100 52,54,55,56,57,58,60,62,65,66,68,69,70,72,74,75,76,77,78,80,82,84,85,
%U A383100 86,87,88,90,91,92,93,94,95,96,98,99,100,102,104,105,106,108
%N A383100 Numbers whose prime indices have no permutation with all equal run-sums.
%C A383100 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.
%e A383100 The prime indices of 18 are {1,2,2}, with permutations (1,2,2), (2,1,2), (2,2,1), with run sums (1,4), (2,1,2), (4,1) respectively, so 18 is in the sequence.
%e A383100 The terms together with their prime indices begin:
%e A383100     6: {1,2}
%e A383100    10: {1,3}
%e A383100    14: {1,4}
%e A383100    15: {2,3}
%e A383100    18: {1,2,2}
%e A383100    20: {1,1,3}
%e A383100    21: {2,4}
%e A383100    22: {1,5}
%e A383100    24: {1,1,1,2}
%e A383100    26: {1,6}
%e A383100    28: {1,1,4}
%e A383100    30: {1,2,3}
%e A383100    33: {2,5}
%e A383100    34: {1,7}
%e A383100    35: {3,4}
%e A383100    38: {1,8}
%e A383100    39: {2,6}
%e A383100    42: {1,2,4}
%e A383100    44: {1,1,5}
%e A383100    45: {2,2,3}
%e A383100    46: {1,9}
%e A383100    50: {1,3,3}
%t A383100 Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]==0&]
%Y A383100 For distinct instead of equal run-sums we appear to have A381636, counted by A381717.
%Y A383100 For run-lengths instead of sums we have A382879, counted by complement of A383013.
%Y A383100 These are the positions of 0 in A382877.
%Y A383100 For more than one choice we have A383015.
%Y A383100 The complement is A383110, counted by A383098.
%Y A383100 Partitions of this type are counted by A383096.
%Y A383100 For a unique choice we have A383099, counted by A383095.
%Y A383100 A056239 adds up prime indices, row sums of A112798.
%Y A383100 A304442 counts partitions with equal run-sums, ranks A353833.
%Y A383100 A353851 counts compositions with equal run-sums, ranks A353848.
%Y A383100 Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
%Y A383100 Cf. A351294, A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383094, A383097.
%K A383100 nonn
%O A383100 1,1
%A A383100 _Gus Wiseman_, Apr 20 2025