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A382877 Number of ways to permute the prime indices of n so that the run-sums are all equal.

%I A382877 #7 Apr 17 2025 23:21:24
%S A382877 1,1,1,1,1,0,1,1,1,0,1,2,1,0,0,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0,
%T A382877 0,1,1,0,0,2,1,0,1,0,0,0,1,1,1,0,0,0,1,0,0,0,0,0,1,0,1,0,2,1,0,0,1,0,
%U A382877 0,0,1,0,1,0,0,0,0,0,1,0,1,0,1,0,0,0,0
%N A382877 Number of ways to permute the prime indices of n so that the run-sums are all equal.
%C A382877 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 A382877 The a(144) = 4 permutations of {1,1,1,1,2,2} are:
%e A382877   (1,1,1,1,2,2)
%e A382877   (1,1,2,1,1,2)
%e A382877   (2,1,1,2,1,1)
%e A382877   (2,2,1,1,1,1)
%e A382877 The a(1728) = 4 permutations are:
%e A382877   (1,1,1,1,1,1,2,2,2)
%e A382877   (1,1,2,1,1,2,1,1,2)
%e A382877   (2,1,1,2,1,1,2,1,1)
%e A382877   (2,2,2,1,1,1,1,1,1)
%t A382877 Table[Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[n]], SameQ@@Total/@Split[#]&]],{n,100}]
%Y A382877 Compositions of this type are counted by A353851, ranked by A353848.
%Y A382877 For run-lengths instead of sums we have A382857 (zeros A382879), distinct A382771.
%Y A382877 For distinct instead of equal run-sums we have A382876, counted by A353850.
%Y A382877 Positions of terms > 1 are A383015.
%Y A382877 Positions of 1 are A383099.
%Y A382877 Positions of 0 are A383100 (complement A383110), counted by A383098.
%Y A382877 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A382877 A056239 adds up prime indices, row sums of A112798.
%Y A382877 A304442 counts compositions with equal run-sums, complement A382076.
%Y A382877 A329739 counts compositions with distinct run-lengths, ranks A351596.
%Y A382877 A353837 counts partitions with distinct run-sums, ranks A353838.
%Y A382877 A353847 gives composition run-sum transformation, for partitions A353832.
%Y A382877 A353932 lists run-sums of standard compositions.
%Y A382877 Cf. A000720, A000961, A001221, A001222, A130091, A329738, A353833, A353852, A354584, A381871.
%K A382877 nonn
%O A382877 1,12
%A A382877 _Gus Wiseman_, Apr 14 2025